Step-up (boost) voltage regulator
Will it work correctly?


Maybe.  There's three parts to your question.


Will the approach work? -- Maybe, maybe not.  There's a possibility that they designed that wall-wart to feed straight into the battery pack with no other management.  This scheme would work if the thing has NiCd or lead-acid batteries.
Are the parts you've selected of sufficient quality?  -- Maybe, maybe not.  I'm going to put aside my prejudices against no-name boost converters bought off the Internet for this one.
Assuming that the answers to 1 and 2 are "yes", here goes:


The way you do these calculations with switching regulators (like your step-up converter) is to remember that power out = power in - losses.  It's safe to assume that even a crappy no-name eBay special converter will be 80% efficient (although there's always surprises out there).  If you do the math, this works out to power in = power out / efficiency.

So, you need 27V @ 85mA, which works out to something a bit over 2W, certainly less than 2.5 (I'm doing the arithmetic in my head -- feel free to check my work).  You have a power brick that'll do 12V @ 1000mA, or 12W.  Even given my in-the-head calculations, you're fine.


  Is it safe?


Probably -- see my prejudices against no-name Internet purchases.  There's a slight possibility (especially if the thing's got a really lame charging system that just goes straight from the wall-wart to the batteries) that you'll toast the batteries.  However, this would Simply Not Be Done with LiPo cells, and NiCd, NiMH or lead-acid cells won't burst into flame -- they'll just quietly die.