if you have a very small amount of bitcoin then claiming these new fork tokens will give you an even smaller amount since they are not worth that much, you should not bother claiming them. for example the last one (i think it was diamond) was worth 0.00017000BTC and if you had something like 0.01BTC claiming it and selling would have given you 0.00000170BTC (170 satoshi)! that is not even worth the bother.

also as you talked about moving bitcoin, since you are moving from your own wallet to your own wallet (from one pocket to another ) you can pay a small fee. for example use 10 satoshi/byte and then use https://pool.viabtc.com/tools/txaccelerator/ accelerator on the hour.