I do not believe you can go higher up without causing more conflict. You have every right to debate this with your line managers - go argue your case - but unless you can show that they have acted completely unreasonably or that this situation is critical to your team's existence, then you have no grounds for going higher up.
There is no way to go above your managers' heads without making it a challenge to their judgement. If there were a way to keep it secret (for example, to arrange for an apparently unrelated external event to require changing things your way), that might work but I wouldn't respect your bosses' managers if they agreed to do that in this context. In this case, it is going to be very clear that any intervention from above was instigated by you. Is it worth it, for this? You will permanently damage your relationship with them.
I think your boss has the right to make the decision she did. If she doesn't like working with that organisation, she is in the position to make that call. It doesn't matter that there's a policy saying it's permissible to work with that group. If there were a policy making it mandatory, that would be different.
Note: I see from your comments above that you hate confrontation and do not deal with it well. But debate (which will only involve confrontation if it becomes unreasonable) is essential here. If you do not debate it with them, you have absolutely no grounds to go above their heads. If I were their boss and you came to me with your objections, I'd say "So what did they say to your argument? Oh, you didn't present it to them?". Not giving them the chance to examine your reasoning and present their own before taking it upstairs is dishonest and unfair. Ironically, it also guarantees the conflict you say you don't like.
You should consider assertiveness training. Assertiveness is all about being prepared to state your case reasonably. Your lack of assertiveness is the problem, here, not your manager's lack of judgement. Until you learn to stand your ground and make your case, your manager will not have the chance to show whether or not she is truly unreasonable.
Go back to them and explain why you feel it is important. Ask them to reconsider. If they still say no, say OK but ask for their reasons. If those seem unjustified and this really is so crucial, then take it upstairs. And be prepared for conflict.