You raised a good point, and I can't find anything definitive. OOTP tries to mirror MLB rules pretty closely but certainly isn't authoritative for such situations - it lets me DFA someone on the 60-day so I'll assume it's legal, but with low confidence.
But if the rules are strict like you suggest, either 1) they will opt to rip the bandaid off, so to speak, very soon and DFA him before ILing him, or 2) they are already on the hook given that he's injured and will have to let him accrue benefits for the whole season. The latter is a relatively minor financial aspect, and service time for a guy in his situation is not really an issue for a team - but the Twins have always had the philosophy, "watch the pennies and the dollars will take care of themselves."
We'll get a refresher course in rules arcana by watching what the Twins do, if no one here speaks up definitively before that.