Lol, when I saw this hand I thought "Villain flopped a weak draw and wants to get there cheap". Ding-Ding-Ding!
Op, I think you intuited the situation perfectly on the flop: you felt that despite the wierd min-raise, top/top was probably still the best hand. Your thought process and plan is flawed, however. It shouldn't be "I want to get to showdown cheap with a decent made hand", it should be "I want to extract max value with the best hand". Flatting is fine, as it keeps the draws in, but is there love for a thinner 3!? Say to $75-80? (I'd just hate to get 4!-bluffed off a decent value hand).
The turn brings an over, but across his range Qx makes but a small portion. There's a decent chance that if we were good on the flop, we're still good. Why should our plan change? Additionally, there's now a flush draw and straight possibilities to worry about. So we should value bet. Villain's flop min-raise sows just enough confusion and doubt, however, we cautiously check to see what he'd do. GG him.
On the river, the board's now super wet, with 3 clubs, cards that hit random Broadway hands hard...NOW is when the plan should change to "Let's see a cheap showdown with a decent made hand".