Good: that'll be helping. If caulerpa does attach to your live rock, a week for that rock lightless in a bucket with a bubbler will do it in. Carbon is good.
I'd query 2 things---number one, accuracy of your tests, which I have to assume are accurate and your nitrates really are that high. That would lead me to suspect you've had a little die-off of your denitrifying bacteria due to the Chemiclean, but it should bounce back shortly---that's why I suggest low feeding, cleaning or gradual removal of all filter media, [filters actually depress bacterial growth in the sand/rock, because they take the food away and don't get any further in the nitrogen cycle than nitrate]. You should have plenty of live rock to do the filtration. I take it you have a 4" sandbed [takes about that much to do proper layering of bacteria]...
Your plan is good: a 20g will give you a lot more oomph, with that 10g backup. I don't know the topfin 10---am thinking it might be a filter, but let me go through the checklist of common problems: filter media should be rinsed out in ro/di water weekly, be it sponge or floss. A hang-on-back [HOB] skimmer would help you a lot: the Remora would definitely work for your size operation, and I'm sure there must be others.
Second thing I'd query: a very surprising source of nitrate can be your fresh water source: some cities in agricultural areas have a lot of nitrate getting in from surrounding farmland. Unless you're using ro/di water [you can get that from a ro filtering station at your local Walmart] you may be picking it up there as fast as your other methods are taking it out. To check that, you need a TDS meter [total dissolved solids]---or ask your lfs to check the local tds; and they may just know the answer if you ask them. Very good your ammonia is staying down: that's the killer. Your carbon is taking that out. Do keep your carbon fresh every week: forgotten carbon can start releasing everything it collected back into your tank when it saturates.
Here's a brief list of what does what:
Macro algae takes out phosphate and other nutrients, helps with water purity: caution on too much in tank, because it absorbs oxygen and gives off carbon dioxide at night, the exact opposite of what it does by day.
Carbon removes ammonia, the most dangerous component of the nitrate/nitrate cycle.
Polypad/polyfilter [changed regularly] removes metals like copper and iron and organic contaminants.
Phosban removes phosphates.
A water change immediately lowers nitrate and other undesirables.
If you ever have a tank calamity, have something fall into your tank, have an algae bloom, etc, do two things fast: run carbon and do a 30% water change, then run polypad.
Emergency supplies: I keep carbon and salt on hand, should keep polypad [haven't had outstanding luck finding it these days], and I also have Amquel at hand in case the chemistry should really go south. I've actually used Amquel in a near tank-crash [prolonged skimmer failure that had gone undetected] with sps corals involved---and the corals and fish were not harmed, so I trust it pretty much. Phosban is not a bad thing to have in your kit, too, not to mention, of course, adequate tests to figure out what's wrong. I have this little chest of supplies and tests that lives right by the tank, and it's saved my bacon no few times.
You can also, if things get really crazy and you don't have a skimmer, aim a pump's outflow into a water chamber like a spare tank and skim off the resultant foam---that's pretty much what a skimmer does, and you just pump the slightly cleaner water back into your tank: it's a makeshift skimmer action that can help in a crisis. My breed of sump does this just in the way it returns water to the tank, so there's always a nasty clump of foam floating over near the inflow, before my skimmer ever gets a crack at it, but the double cleaning action doesn't hurt anything, and the regular skimmer will get it sooner or later.
Anyway, I hope this is of some help...I just got THROUGH the "move" thing, and know where you're at: everything's "make it through to the new system"---mine got dicey when I planned on a move on the 15th and the thing got delayed to the 29th and then my sump didn't come through to let me finish the plumbing---quite a wild and wooly operation....you wouldn't believe what a hassle it is trying to reduce a pipe/hose by 1/4 inch to make a necessary connection! I'm sure there are specialty connectors out there, but it was a hairy problem at the stores I frequent.