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.