Ok, sorry, I mis-read part of your description.
If you
only have balanced coming out of the preamp at +4dBm and
only have unbalanced going into the deck at -10dBV (you may want to double-check your deck specs on that; while not impossible, -10dBV unbalanced XLR inputs are a bit rare), then you'll either want to use a matching transformer or use your mixer for that function as you originally questioned. While it is sometimes possible to get away with running the balanced out into the unbalanced in if you have an input trim/level control robust enough to handle it (and the pin outs on the unbalanced XLR are correctly wired), I wouldn't recommend it. In that case, yeah, use the mixer in-between to make the step for you.
An alternative is that you can get an in-line transformer that does the balancing act for you instead of using the whole mixer. But those would cost another $50 bucks or so per line. If your 1604 sounds clean enough for you that you don't mind having it in-line, use that and save the $50 for cleaning/demaging your tape deck or for some fresh tape for it.
G.
EDIT: If you have the original 1604, I make no guarantees about the about the balanced/unbalanced out capability I attributed to the 1/4"ers on the VLZ. I don't remember the differences between them well enough. You might want to check the specs on the original 1604 in that regard if that's what you have. In a worse case scenario, you could alwyas use the RCA tape outs on any of the 1604 models, which are all definitely unbalanced.