You don't need to treat the inverts and nem. Just leave them in the MD. Feed sparingly once or twice a week to keep the cycle going and feed the invert life still in the tank.
I was like you when I first started in saltwater. I tried other methods first to eliminate ick from my MD, but none of them worked. If you are serious about getting rid of ick from your MD , follow one of the 3 proven methods to eliminate it once and for all. The truth is, some people are happy just managing ick rather than eradicating it from their MD, and they are comfortable with that choice. It works for them. Some are more successful managing than others. I hate losing even one fish. You'll have to decide what works for you. I got sick of ick coming back anytime I added a new fish, a power outage occurred, or some other stressful even such as a PH drop happened. If you're in this hobby for long enough, you'll soon learn that
#$%^ happens to the best well-kept tanks at some point. If a problem can be solved, not managed, I'm going to solve it...but that's just me. Keep in mind that if ick is in your tank, you have two fish that are known ick magnets.
With that said, I wouldn't put the two larger fish in anything less than a 20 gal. a 29 or 30 gal would be better. Because you have 9 fish, I would shoot for at least a 40 gal tank if you can swing it. If not, nothing less than a 30 gal and I would put in a LOT of different sized pvc for them to hide in as they would definitely be overcrowded and you don't want any bickering. I'm lucky. I have a 75 gal tank to treat my whole 180 gal family of fish if needed, but even that was still too cramped for the size fish I have. The good news is that if you can keep them from fighting with a lot of pvc and keep water quality from going south from the over crowding, it is a temporary housing so it's probably just fine for the short time you'll have them together. You can always grab a cheap 10 gal or purchase a divider and section off a fish or two if some are so stressed they stop eating or just won't play nice.
The personality of your fish will also dictate how big the tank should be. For example, if your puffer feels cramped, it could turn on the smaller fish and take a chunk out of one. My 6-7 inch porcupine puffer bit the head off my flame hawkfish in a 75 gal fish tank. My Toby puffer preferred to stay hidden in a large PVC pipe and only came out when I fed the tank. Tangs are often active swimmers and need room to move around...but again, personality plays a part. I have a yellow eyed kole and it pretty much sticks to a rock formation that has multiple holes to swim around in my 180 gal tank and it does that instead of free swimming the length of the tank. The smaller fish you mentioned probably don't need as much room as the PBT or puffer, just appropriate sized pvc pipes to hide in. Others here may have different suggestions. This is just what I would do in your circumstances. Good luck!