Ok, in that case it would only make sense to close out the account if it were going to stay closed for at least 10 months (actually more like 11 or 12, since avoiding the paperwork is worth a couple of bucks) -- but by then it would be time for you to reopen it, so you might as well leave it alone and eat the $2.50/month.
I do still think you should "back-load" your 2016 contributions and front-load your 2017 contributions so as to go from $0 to $5000 balance as quickly as possible and minimize the duration of the $3/month investing fee (or the opportunity cost of uninvested cash). During the months that you're working on that, you should pay the minimums on the loans. (Please don't miss out on maxing your 2016 HSA; let your "pay off debt by the end of the year" goal slip if it comes to it.)
By the way, I realize you've already decided to pay off your debt first, but if the interest rate on it is 5% or less -- or if you're in a high tax bracket -- I think you should reconsider. Personally, I'm maxing my 401k, HSA, and IRAs for myself and my wife and only paying extra on my student loans with what's left over after that, and it's working out very well for me. The tax savings* alone from dropping my AGI compensate for the student loan interest (which itself is deductible, LOL), so the investment returns are just icing on the cake.
(* Technically, that tax is only deferred, not "saved." However, by the time I pay it I'll be retired, which means that either (a) I'll only be claiming enough earned income to meet my modest expenses, which means I'll be in a very low tax bracket, or (b) my investments will have done so well that I'm drowning in money, which means I won't really care how much tax I'm paying.)