Given an infinite resources, the value could be fixed such that this doesn't affect Bitcoin's stability. The value can't be fixed to respond to changes in the exchange rate vs fiat but it could only be fixed to a certain Bitcoin amount. It would be hard for the network to reach a consensus about the value of Bitcoin unless there is a central authority (exchange, weighted average of exchanges) to decide this.

Bitcoin's realistic minimum fee is 1 satoshi/byte.

Assuming that you're talking about minimum fees for a transaction to get confirmed within a timeframe, it shouldn't be fixed. Bitcoin's capacity is not infinite and will probably never be. The fees market won't be affected much even if you were to impose such a restriction. The miners can only include that many transactions and not all the transactions above the fee rates would be cleared in the timeframe; the restriction would be pretty useless.