The distinction between slow and fast film has nearly vanished; those explained in old books can be largely ignored.(posted 8062 days ago)I don't recommend HC-110. I know many people who have HC-110 halfway used and never finished because they found D-76 better, or started mixing their own chemicals (or take a couple bottles off my mixed chemical shelf). HC-110 is too much of a compromise unless you are shooting LF, in which case you can afford such a compromise.
Drop the idea of pushing for now. You'll regret when your eyes become more critical months later.
T-MAX developer, I think, was originally designed to compensate for lack of shadow details with old T-MAX films. I've read on a British Professional Photographer article that some author there complained Ilford about lack of shadow details with Delta films, and Ilford secretly told that guy to use T-MAX developer. As EKC silently improved the emulsion and thereby eliminated the special need for T-MAX developer, they started advertising T-MAX developer as a push developer. Ilford, on the other hand, put "Professional" designation when they improved it. T-MAX developer might be a good choice for Delta 3200 and TMZ, but then why not Microphen?
My recommendation is the same as John Hick's recommendation. I'd use Microphen instead of DD-X, partly because Microphen is cheaper and its (unofficial but very close) formula is published (should Ilford change it or discontinue it I can still make some and use it). But this is a very small point -- DD-X or T-MAX dev is probably just as good.