Monks at a Carmarthenshire multi-faith community say they are "pretty much at the end of the road" in their legal fight to save Shambo the bullock.
There has been a high-profile campaign to save Shambo
The Welsh Assembly Government has also ruled out the possibility of another skin test on the animal.
The Friesian at Skanda Vale, which has tested positive for bovine TB, is again facing slaughter after an appeal court overturned a ruling to block the order.
Brother Alex described the chance of an appeal to the House of Lords as "slim".
The appeal court's ruling has been welcomed by farmers' union NFU Cymru.
The assembly Government won its appeal against a high court judge's decision last week to block two slaughter orders. It said there was no timetable for slaughter, but it wanted to minimise distress to the animal and its carers.
Judge Gary Hickinbottom had said the orders had failed to give enough weight to the rights of the monks, but the assembly government's appeal was upheld by the appeal court in London on Monday.
Lord Justice Pill said former Rural Affairs Minister Jane Davidson acted lawfully when she refused to make an exception for Shambo as a sacred bullock.
He said the decision was justified even though Shambo's slaughter would be considered a sacrilegious act by the community.
Brother Alex, a senior monk, told BBC Radio Wales the community had faced the problem of trying to convince the courts that "untested and untried" methods of dealing with possible bovine TB would be effective.
He said: "There is a small possibility we could petition the House of Lords but just in logistical terms, never mind legal terms, it's quite a slim chance.
Shambo the bullock is being kept in a special enclosure
"So it looks as though we're pretty much at the end of the road in legal terms."
However, any possibility of the monks petitioning the Lords seems to have been made impossible by the appeal court.
The judges refused their appeal for a seven-day stay and in order to appeal to the Lords, the monks would have had to return to the High Court to extend the stay until October.
Brother Alex said Shambo's fate had "touched a raw nerve among the Hindu community" and he called on the assembly government to be "magnanimous" and provide one more skin test, even though he accepted there was only a remote possibility it would be negative.
He added: "It would give a little bit of confidence that they did understand the sensitivity that was involved in this one."
But the Welsh Assembly Government has rejected the possibility of a re-test.
A spokesman said: "The skin test is the best test available and it is designated by EU legislation. We do not believe that a re-test is necessary or appropriate."
The community has fought a campaign since a TB test returned positive in April, saying they were guarding against Shambo infecting other animals.
But they have faced opposition from many quarters, including the farming sector, where the animal is seen as a TB risk to livestock.
Dai Davies, president of NFU Cymru, said the sooner Shambo was slaughtered the better.
He said: "The longer this animal lives, the longer the [bovine TB] reservoir will be in the heartland of livestock-producing areas of Wales.
"The sooner is it done, the sooner that reservoir of disease is removed from that locality."
The assembly government said it was looking to move forward, but it was not possible to give any timetable.
"We are glad that the court has accepted the important public and animal health arguments in this case," said a spokesman.