This would be better served on a finance forum. Having said that I will give you a direction to start with.

You can search for all the "free" ($0 fee/0%) BT offers. (Chase Slate, Discover, Cap One, etc)

If you run out of the no fee offers and cant cover the whole thing I would consider leveraging any equity in your car. You can refi your car for more years up to it's current value and cash out the rest (Penfed 1.99% or less comes to mind).

All of this is considered extremely risky especially since you are "planning" 10-12 which could easily turn into more given another unexpected expense. I wouldn't do it and will not go into more detail but this doesn't seem like you have a travel angle (no points worth having for BT's anyways) so I would stick with financial forums and consider all your options.