There are very few contractors that know enough about or are still willing to repair radiant heat, and it is often a very expensive and time consuming repair. It is usually easier and more cost effective to install an alternative form of heating.
If the heat is not working in only one room I doubt the lender will get in the way of a closing. Heating systems need to work in order to close escrow.
You should find out if the seller still carries their home warranty. If they do, the home warranty company should cover the repair or install a new heating system. That is what the insurance is for.
If this is the only heat in the home, then yes, the seller should either repair or install a new source of heat.
I sold a historic home (late 1800's) with NO heating system. It had a centrally located fireplace which actually heated up the entire 2 story Victorian home very nicely. The lender actually approved that one. Actually I sold that particular home twice.
If it does not work, then you need to have your agent go back and negotiate to have it repaired/replaced or credit you for an amount you both agree on.
All the best!