The increase in price is a factor that affect the number of bitcoin transaction. The higher the number of transaction, the denser the network will be so that the transaction costs will be higher. So in my opinion, it is not the price increase that causes high transaction cost but the network congestion.
I can't be sure how much it will cost in future Bitcoin transaction because it's very difficult. Miners will face hard time as some of them will have to stop activities that are no longer profitable. But I believe the transaction fee will not exceed the reasonable limit.
I think it is complementary when the network has congestion, the price will increase as well. The price will increase because there are people who will pay for high tx just to have confirmations to the transaction. And since the network is congested, people who are in rush will obliged to pay for the high tx. The mining difficulty will also be a factor to consider when the price goes up. As we can see, when the supply was halved the price of bitcoin moves up as well. In the future when the supply will be mined, miners will dictate the tx fee and for sure it will goes up.