Linking apartment life base lot with the sub lot?
I have no answer to your question, btw - I did try using the staythingsshrub once and it royally messed things up. The apartment whose furniture I wanted to keep lost it anyway, and things were messed up for the other family (the "base" family - first to move in) in the building too. I don't remember the details of what was messed up for them but I ended up moving them out and back into a newly-placed clean copy of the same lot, which luckily was fully furnished and decorated as I wanted.
Before that failed stayshrub attempt I had always used a house sitter sim to keep apartment furniture. I'd move a new sim into the apartment in question, joining the family that lives there. Then I'd move the family out and keep the house sitter sim in the apartment to keep the furniture. Then once I had a new family ready to move in, they'd join the house sitter, and then the house sitter would move out. Or you could skip the house sitter step and combine the families directly if you already have a new one lined up. Of course moving sims out this way they lose their accumulated clothing, which I find super annoying - but worth the price if you want to keep the place furnished as it is.
Hmm, maybe "realtor" would be a better name for the house sitter actually. Randy Realtor. There you go.
Just adding I mean after a sim has used them, not fresh placed from the catalog.