Mr Sheahan told the meeting that it was highly unlikely that the European legislation would ever be changed and certainly not in the short to medium term.
"This is a rule that was agreed by 28 member states and took five years to be agreed on," he told the meeting. "Like it or not, we are stuck with it."
While no-one could argue with the food safety reasons behind excluding such animals from the human food chain, there have been calls for horse meat to be used for pet food.
However, EU rules also exclude meat from those horses from commercial pet food manufacture either.
"There might be a very small chance that this could be changed but even if it was, it would only be a minor help.
"The pet food market for horses will not solve the problem of 16,000 unwanted horses this year," maintained Mr Sheahan.
Pat Hayes, operator of Ossory Meats, one of only two remaining factory outlets for horses in Ireland, told the meeting he was willing to offer lairage facilities so that horses could be blood-tested for bute drugs prior to slaughter.
However, Mr Sheahan said that individual blood testing for bute would cost in the order of €60 per animal and testing would also be required for other drugs such as sedalin, again at €60 per test.
On a cost basis only, this would almost certainly rule out the possibility of testing horses prior to slaughter but in any event, horses that have been administered any of these prohibited drugs would still not be allowed into the human food chain.
"Unfortunately, it is a bit of a non-runner," said Mr Sheahan.
He insisted that the only disposal option for the thousands of unwanted horses without passports or with unsuitable passports was the local knackery facility.
Knackery disposal of horses has increased in recent years, but not as dramatically as the slaughter option.
In 2008, some 1,800 equines were disposed of through knackeries, rising to 2,600 in 2009 and 3,000 in 2010.
Exact figures for 2011 and 2012 are not available but Mr Sheahan said they showed similar modest increases.
However, animal welfare groups claim that irresponsible owners will not pay for disposal of their horses through the knackery system.
"They will simply dump the horses on public land or in a local forestry to starve to death," claimed Barbara Bent of the ISPCA.
"Until the identification of horses regulations, complete with transfer of ownership, are implemented, we have no way of pursuing these people and they will continue to get away with it."
Cases the ISPCA has dealt with recently include a band of eight broodmares with foals at foot and in foal again who were abandoned in a wood last October.
"By the time we got to them in February, quite a few of those had already died," Ms Bent told the meeting in Abbeyleix. "We had another emaciated thoroughbred mare who had been clipped out and hunted, then dumped in a bog."
To date in 2013, the main problems with horses monitored by the ISPCA have been starvation, lack of dental and hoof care, injuries from sulky racing and unwanted horses dumped in remote sites and on public land.
Worryingly, calls to the ISPCA national helpline are on the increase in 2013, with a 33pc rise in the number of calls in the first five months of the year (see Table 1).
"I'm afraid this autumn and winter are going to be the worst we've ever seen for starving horses because fodder is in short supply and will be very expensive," warned Ms Bent.
The ISPCA woman called on the Department of Agriculture to provide a disposal facility to take horses with no future.
"That is an unusual thing for an animal welfare organisation to say but these animals have to go somewhere and we would prefer they were disposed of humanely than left to starve to death in forestry somewhere," she said.
Ms Bent called on the government to introduce an initiative to castrate colts in order to prevent indiscriminate breeding among large groups of low quality animals.
"Overbreeding is one of the biggest causes of unwanted horses in 2013," she told the meeting.
"The lack of castration means that horses are breeding with no market and no purpose for them.
"We have been dealing with one man who three years ago had 22 horses. He left the colts uncastrated and he now has 80 horses on the same property," she said.
"Those horses are of no quality and they have no future."
Ms Bent urged the entire equine industry, from individual breeders right up to the Government, to pull together and take collective responsibility.
"The solutions to this crisis are to reduce the number of horses being bred, enforcement of the identification regulations and the control of horses, as well as good strong bylaws to tackle the unregulated and urban horse sector," she insisted.
Table 3 shows the number of horses seized by local authorities under the Control of Horses Act and the higher proportion of those horses that were humanely destroyed after seizure.
Horse Sport Ireland chairman Professor Pat Wall suggested that European farmers organisation COPA and MEPs could be helpful in lobbying for a change to the existing rules on animals entering the food chain.
"It's understandable that Minister for Agriculture Simon Coveney would not want to be seen to be lobbying for more horses to go into the food chain. He's at the head of a country that exports €8bn of top quality food and drink annually," remarked Prof Wall.
"But perhaps the European farming organisations and MEPs could gather European colleagues to put pressure on the Commission to change the rule.
"A derogation that would allow owners to test animals for drug residue prior to slaughter might be a long-term solution but we need all solutions, short, medium and long," he told breeders.
"A welfare crisis this autumn is going to cost the Government millions and create awful, negative headlines," he maintained. "If that's the case, maybe the money would be better spent now to prevent that negativity."
Prof Wall was among a number of speakers at the meeting who commented on the possibility of a scrappage scheme for unwanted horses.
"The mention of a cull or scrappage scheme would send out a very negative image for Ireland but maybe there is scope for an assisted disposal scheme to take these horses out of the system," he said.
"Could the money that the Government will have to spend dealing with a welfare crisis be used wisely to address the problem in a constructive way without adverse publicity?" he asked.
However, Department of Agriculture officials have poured cold water on the prospect of any disposal scheme being introduced.
Privately, officials have said there would be no justification for paying people to dispose of horses that were being kept illegally.
They added that in the face of cuts to specials needs assistants in schools and healthcare, the idea of providing funding for a horse disposal scheme would be abhorrent to politicians and the general public.
Speaking at the meeting Mr Sheahan said: "Any scrappage scheme would be a political issue, but not something many politicians would want to be associated with."
Prof Wall said the Government and industry needed to explore all avenues.
"Could LEADER or Pillar II funds in the CAP be used to create outlets for some horses?" he asked.
"Are there opportunities to market some of the horses as riding cobs and family cobs or for eco-tourism?"
However, Prof Wall was adamant that individual breeders needed to take responsibility for their actions.
"I have two mares myself and every year I think the foals will jump on the Aga Khan team," he told the crowd.
"I am a dreamer, all horse people are dreamers but there are some people who are hallucinating.
"That's being a bit facetious about what is a very serious issue.
"We have to realise that there are two sides to the selection coin.
"The first is that you select and breed from the best mares you have but the other is that you must cull the rest."
He reminded breeders that in order to avoid another crisis next year, they should ensure all 2013 foals are registered on time so that they are ruled not out of the food chain.