I'd like to ask if I have to transform right after doing Gibson Assembly or if it's okay to store the Gibson Assembly product in -20c (or 4c?) and do transformation later (~a week later).
This is not a direct answer to your question, but you should give it a try without adding the gibson mix, the bacteria can do it almost as efficiently by themself:
Article In vivo DNA assembly using common laboratory bacteria: A re-...
it should work fine, I stored some of mine (-20) for up to a month.
I had issues with the Gibson mix, tho. It seems it has an incompatibility with some E. coli strains. Perhaps you could use those PCR purification kits if you see any decrease on transformation efficiency.