In the scenario given, you don't have the prefect solution to drink fresh coffee. However, I would opt for carrying a brewing solution (French-press in your case) with me. Let's discuss the two cases separately.
In that case, you can grind fresh beans and prepare your coffee optimally with the brewing method of your preference. However, you cannot protect the brewed cup of coffee to go 'bad'. The reason is, whatever you do, you leave tiny coffee grounds in the water and they continue the extraction process. By time, your coffee gets bitter as a result of over-extraction.
Actually, this is a problem even in our daily routine. Pour-over brewing methods are invented for that. We mostly use paper filters to filter out most of the tiny grounds and have a clean cup to get rid of continuing extraction. Thus, we can have a stable cup while we drink in the next five minutes.
If you follow your taste buds, you can clearly feel this in a Turkish coffee cup or French-press, as most of the tiny grounds are still in the final cup in a French-press (and all of them are still there in a Turkish fincan) you should be quick to drink it. Otherwise, the cup goes crap. (By the way, inverted French-press is can also be used to minimize the effect of tiny grounds. But not effective enough.)
Everybody on this board could admit that after grinding coffee goes stale in a few minutes. (See How quickly does coffee get stale?)
When you grind the beans, the surface area increases so the delicious aromatics evaporate more easily. You probably will end up with a duller coffee experience at the end. However, I feel like over-extraction is a far worse problem in your case.