Warning - very long post ahead 
In my opinion - there are a couple of issues with this hand.
The TLDR version is - I think your post flop actions were reasonable and that may have been the way I would approach the hand, with some conditions and exceptions elaborated below.
1) preflop - I think the problems began preflop - you let the BB do whatever he wants.
I wonder if you considered your raise a bit suspicious. 2.5x is already on the small side (most good players suggest between 3-4x, with 2.5x being possible sometimes), but for blind wars, I think 2.5x is plain incorrect. The reason:
during blind wars, a 2.5x raise means the BB needs lower than usual equity against your range to call. AND he has position on top of that. which means he can play a LOT more hands than normal profitably than he would be able to normally.
you say he's a fish and therefore he will probably not be exploiting that, but in fact, just by giving him better odds, some of his fishy calls become profitable ones.
It also makes it much harder to put players on a hand range post flop. Against a more experienced player, your reverse implied odds would be really big here.
However, I will assume he is not expanding his range further, and just calls a normal "vs. steal" range in position (which is still quite wide).
He will easily call with pocket pairs, some high suited aces, perhaps higher offsuit aces, some high suited kings, some of the higher suited connectors. That is actually not even a fishy range for a blind vs. blind open raise and call, since the small blind's range should be quite large.
His range, even if he wasn't fishy, against a small blind open raise should be wide. You didn't mention this, but I'm assuming he doesn't like to 3-bet. That's important, because if he does, you can remove most of the top end of his range once he calls (however, passive fishy players often don't do that).
2) Flop comes - Surprisingly - against the range I've described, you're only ahead by about 55%-45% (According to equilab. I actually guessed your situation to be even a bit better than that before checking).
There are still quite a few king hands in his range, and many of the suited (and perhaps offsuit connectors) in his range got some straight/flush/trip draws
Still, you're obviously ahead of his range. Betting for protection here, seeing that at the very least the flush draws will call you (and perhaps some other weak pairs).
However, does this player call a lot? or simply raises a lot more postflop than preflop? You didn't mention this, because it's here that you already need a plan.
If this player calls a lot and will call ace highs and weak draws as well on this flop (which his range contains), then you already need to decide which cards on the turn will make you fold and which won't. Otherwise, you would just be folding to a bet if an ace comes on the turn, if any heart comes, if a queen comes, and will not know what to do when you face large bets if a 10 comes etc.
If you think he doesn't get to the turn with ace highs with this call on the flop, your play seems correct to me.
his fold to Cbet is on the low side, meaning he stays in the hand with air quite often.
That is the reason why mid and low pairs are considered horrendous hands when it comes to post flop playability.
So many cards will make you not know where you stand. And playing it out of position for more than just "set value" is quite difficult, because you will, in the majority of the time, face more "higher" cards on the turn or river. But these situations are hard to avoid sometimes
out of position, avoiding the sticky situation and just check-folding the flop ("set mining" with your pair) is possible if you don't like making big hero-calls or be in these sticky situations.
3) turn comes - One of the worst cards for you to see (10h might be a lot worse, since it hits even more of his range). And at this point, if I remove the total nothing hands from his range, even if I leave in the wishful-thinking draws in, according to equilab, you are already behind his range.
In the end, I think I might have played this hand post flop exactly like you did.
Because your other idea - 2nd barreling the turn - is a complicated proposition - if your opponent is a real calling station the vast majority of the time, you might even be able to bet for value here (since the flop call didn't mean too much). Otherwise, if not, even many fish can't really call a 2nd barrel here without a flush or being able to beat one, so 2nd barreling is turning your hand into a bluff (you are basically representing a flush on this board with it, and are expecting overpairs and Kx hands to fold).
Problem is, you are out of position, and only the BB knows if he wants to play for stacks or not... So it'll be a costly bluff.
So the TLDR is - I think your post flop actions were reasonable.
I think with such a player, if he's a bit on the passive side, it's a good idea to give up on the flush card on the turn once he shows aggression.
More aggressive players might just be raising to take advantage of your check after flop cont betting. But still, it's not an easy situation.
In my opinion, if you feel like you're leaking chips in these types of situations, it's perfectly fine to play an 88 as a - "hit a set on the flop or give up", if the stack sizes are big enough, definitely when out of position. You might give up on a bit of profit, but you'll be avoiding huge losses with sticky hero calls you'll have to make for it.
Sorry for the (super) long post
8
6
K
As you said the standard opening size is 3-4x depends on player and position and from the blinds (for me from btn also) it is lower. As we both know villains range is pretty wide here. I think cbetting flop is almost mandatory because: 1) villain at NL10 rush poker will often fold too much to cbets, 2) we are often ahead of his range and 3) check-folding pocket pairs makes us exploitable. If you think x/f is good with pocket pair, isn't it good with a FD also? We have lower equity with FD vs villains range than with 88. If we x/f everything else but Kx hands on this flop what happens?
