Actually, I would recommend getting an Emperor 280. That would be perfect for the 20 gallon tank. The biowheel from the 150 will not fit, for it is smaller than any other biowheel out there (I think - I do not know all the newer, small filters in that line, but you need something bigger, anyway). Since the 150 (or am I thinking of the 110???

: ) runs almost totally on the biowheel (no other place for the beneficial bacteria to reside - except for what little is on the carbon pieces and floss of the pad - that gets grossly filthy and needs washing all the time ANYWAY!), you will need to seed in another fashion.
You have a few choices. That pad is the same size as the ones that go in the 280 - in width. It is too short for the filter. This will hurt nothing. I would suggest that you fill the gray clamshell with media (sintered glass, etc). Then slide in a cartridge in the front slot that is made for the 280. You will now have a tight space between the two that will hold the old cartridge of the 150 - sliding it in tightly right behind the 280 cartridge. It will fit.
I actually have peeled the blue floss of cartridge frames, sewed netting over them and filled them with sintered glass media. These, as extra media, slide into place, too. They work well. You will be doing much the same thing with the dual cartridges - except that any beneficial bacteria you are attempting to salvage is on the blue floss.....
You can then choose to run the 150 on the tank, also, so the seeded biowheel will work for you while the newer biowheel and newer media seeds. Then pull it when you wish. OR ..... you can simply float the biowheel in the tank. I have found that in many tanks, I can "balance" a biowheel under the outflow of a filter, where it will spin in the water, held trapped by the currents.
Your cycle may bump a bit - the biowheel will not be "the same", but I do not think you should see anything dramatic at all - and if you see anything, it will be very short lived.