Apparently, in your company there are at least two teams. Your team needs some functionality from application, and the other team is responsible to deliver it.
Priority is just a meaningless category to prioritize work globally. You can set your own priorities as needed inside your team.
You tried to ask for help from other team. Seems that help is not coming. You need to ask your own developers, how hard it will be to find out where bug is, how to fix it, and if they will be allowed to patch the fix (or some process rules will prevent that).
If your team can fix the bug easily, do so. Doing so you will help the other team which seems to be swamped. Tell them you are doing it to help them.
If your team cannot fix it easily (will take too long to understand the area to make fix safe, and other team already understands that area well, so it will not be efficient use of your team time), manager of your team needs to talk to manager of the other team. Maybe your team can do something else to help the other team to get time to fix your bug.
If all else fails, both managers have boss, and your manager can escalate it up.
If your bug is crucial to you, but irrelevant to users, it might be that it will not be fixed anytime soon. Develop workaround and checklist to contain it.
P0 or not, it is all about triage and fixing most important bugs first, and finding who has the capacity to fix the bug right way.