I would be most cautious about removing close to 50% of the primary bio-media from any going tank. At least move only a wheel and not a whole filter as well. While you do have bacteria spread out all over the hard surfaces of a tank, you still normally have the biggest share in the filter media and the filter itself. And for sure you will still need to monitor the tank from which you take only a bio-wheel. The risk will be greatest in the tank with the cray on eggs if it remains where it is. I do not know the risks associated with moving a cray on eggs as I have never kept them.
If you use a bio-wheel as a jump start to fishlessly cycle the new 60 it should go pretty fast dosing ammonia. I personally prefer using ammonium chloride for this, but it can be done with ammonium hydroxide as well. The problem is if you move the cray on eggs over you can not dose ammonia into the tank. So you can't get the bacteria to multiply that way.
Now I am sure I am going to hear it from the usual suspects for suggesting this, but I know it works having used it twice. Some folks will claim it doesn't. But you can get your new tank cycled using either Tetra Safe Start or Dr Tims One and Only. These are bottled bacteria which should work fine. I used the Dr. Tims, but he basically "invented" both products.
If you are going to go this route please go to the site for either product you choose and read the additional info there. Then follow their directions to the exclusion of what else you might be told here for using them ( you will have no way to separate the good advice from the mostly bad that comes when using these products. If you are not willing to do that, then do not even try this. This route should get the new tank running pretty fast and would then let you move the cray in shortly and the new fish can be added as well. They go in after you get the tank to snuff dosing some ammonia and the numbers are in order.
I am not quite sure what you mean by using chemicals. Dosing ammonia to cycle a tank or build up bacterial colonies has been around for over 15 years now. The bottled bacteria are not chemicals, they are live bacteria.
Finally, bear in mind if you are willing to stock your new tank over time, then the amount of cycling or ready to go bacteria needed are much less than if your goals is to stock fully at a single go.
