First, make sure you carry all relevant paperwork, physically. So, if the company has an agreement with the hotel, carry a copy, if there is an email where they agreed to accept that card, print it, etc.
That by itself may solve it in some cases. A customer saying “my company agreed with the hotel X” is not the same as a document with the chain letterhead, sent by the main branch, saying X.
(Also, it may help clarify the position of both, as there may be fine print in the X-with-company-Y that neither you nor the clerk knew about)
Still, it may be the case that they insist that accepting your employer credit card is “impossible” or even “illegal”. Make them document it. Ask them to provide you a form stating that you asked them to accept card 1234, according to document XYZ, but that they can’t do it because <their excuse>, there’s no other option than using your personal credit card and that they accept that if it’s an error on their part, they will be accountable for any or extra costs caused by that. Have it signed by the employee, and ideally also with the hotel stamp.
Preferably, prepare the text in advance and review with your legal department, and have such document prepared with the relevant legalese and a space where they can fill a description of the issue.
If it is really impossible to do otherwise, there should be no problem to acknowledge that in writing, right?
Well, you will probably find out that before doing so they will try extra hard to do it. Or suddenly, there’s actually another alternative.
If they refuse, argue that you need to justify such irregularity (after all, why would your company reimburse you if they already handled everything?). If they do, give them your personal card and continue. Later at the office your accountants can review the stated problem and decide the best course of action. It is possible that some part of the agreement with the hotel was unclear, and they need to add extra clauses on next booking. It is possible that the hotel is at fault, in which case you have both a document where they agree to accept a card and another where they refuse to do so, which seems a pretty clear case for requesting them to reimburse your company the taxes, the cost of the work of your accountants processing your reimburse, the consequences if your card was compromised due to a breach… Or they may prefer to not pursue action about that (yet), simply complain to the hotel liaison that agreed to accept the card (which should make the hotel aware of the problem, eg. it may make them teach their clerks about the proper procedure on these cases), or even do nothing. In any case, you are forcing the hotel to state their contradiction, and should be the starting point.