The price going up won't result in the fees going up.  The only thing that'll result in fees going up is an increased transaction volume and total size of transactions being added to the blockchain.  If the price goes up, mining revenues will rise because the number of Bitcoin they get as a block reward is set - if the block reward is still 12.5 and the price doubles, the revenues will double, meaning that miners no longer need to receive as much of their revenue in transaction fees.

Bitcoin's fees won't actually get that bad because the transaction volume will decrease as investors store larger amounts of money in it because of the bad fees, making Bitcoin accessible to less people (thus decreasing the transaction volume, but potentially keeping the price the same or even raising it).  So if Bitcoin isn't scaled, there will be a middle ground in which transaction fees are high but not unbearable.