The fact that ETH fees are high at the moment doesn't make BSC any better. I can tell you the BSC network also would get congested with high increase in fees too if it gets to adopt like Ethereum blockchain.
The cost on BSC will be high if the price of BNB has gone up by around $1000, because what is cut when making transactions on the BSC network is BNB and nothing else so this is something that everyone should be aware of at this time so that when the fees feel high they don't make them users BSC became complaining about it.