The flop is so obviously better for his range than yours that I would start off by checking and probably folding to a bet.

It is true that this flop is better for his range than yours, but that does not necessarily mean that you should start by checking. If we check, we are checking to give up. Since that is essentially a 0 EV play, if a bet is profitable, than we should bet. Against a tough opponent who will recognize that the board is better for his range and appropriately attack your bets by floating wide or light semi-bluff raising, you need to give up. However, your bet here only needs to work ~40% of the time to be profitable (and that ignores that we have some EQ/brrl opportunities when called, meaning your bet probably only needs to work ~33%) and there will be many opponents who will fold quite a bit more than that. Therefore, if your opponent is on the tighter side, you can still profitably CB here despite the board disadvantage.

This turn is terrible for your hand and great for villain's range. I would not expect to have much Fold EQ with a bet and your hand does not have enough EQ against villain's strengthened range on this turn to bet profitably.