Cutting the biowheel would work, but they tend to be rather expensive to replace and I am assuming you want to use this filter down the road.
You have several options - depending on how soon you wish to move the betta in.....
You can stuff the sponge from the Aquaclear into the back of the 150 - remove the cartridge if you wish - it is not important at this time..... STuff that sponge in there and it will seed quickly.
If you want to run the 150 for a bit with the betta, you can wrap a sponge around the intake and slow it down, perhaps..... maybe. I do not know how little a betta needs.....
You can go ahead and move your goldie, and put the Aquaclear on the tank. Pull the biowheel off. Either float it in the betta tank or I like to run string on them tieing it on either end. I then suspend it beneath the overflow of the new filter so that all the water runs over the biowheel. This will help process the betta's waste.
It does take a while for the beneficial bacteria to transfer from one source to another if they have to travel through the water - something they are rather loathe to do - particularly if there is less waste load than before and the bacteria colonies are not blooming. Any contact that you can make between the biowheel and the sponge filter the better. In a larger Aquaclear, you can jam the biowheel down sideways into the filter. I took the pleated material off one once and stuffing down into a filter in contact with the media for that filter. IT was a solid cycle within 2 weeks for full goldie waste. The pleated material is a bit fussy to get back on - but with tweezers and some time, you can do it just fine.....