In the situation you described I don't think bluffing is a bad thing, but I do think your amount is extreme and you put yourself into a tricky situation unnecessarily. So let's look at the hand and then things you did well and then the things I think could be improved on.
You didn't mention on how many players are at the table so I'll just assume a full 9 handed game. CO late position raise with the K♦7♦, totally fine with the action. Bet sizing can depend on the table but something like 2.5x to 4x is fine, maybe even higher on some cash tables.
So the flop comes, pretty good flop given our hand, T♦9♥Q♦, so I think given our hand it make senses to bet it, especially after the check is to us. Helps us build the pot and maybe just win it without showdown. The 2$ is a bit on the larger size into a pot of $2.10. I think realistically you can essentially achieve the same thing with a 40% to 70% pot sized bet. I'll give you the benefit of doubt and assume the player was calling wide so build the pot, however if that is also true these type of players tend to not like to fold when they hit a piece.
Turn is a T♠. Not great for our hand obviously and a paired board can sometimes be tricky and you really need to understand your opponents range here. So at this point I think the pot is $6.10, although you did say $5.80 was pot but not sure what happened to the small blind then pre-flop (maybe raked already), in any case it's not a massive difference. I think after your opponent checks to you on this turn and you pot it, you should ask yourself before you pot here the $5.80 what is that saying about your hand or trying to achieve? Put it this way this type of bet is very polarising, like would a made hand that wants to get paid bet like this? Would your opponent even be thinking like this? Essentially it's one of those situations where the person at the other end, if they have something may think to themselves if they have it fine, but it's either I'm crushed or I'm good. I do think the $5.80 pot bet is massively inflating the pot here that puts you in a very difficult situation on the river. Personally I'd have checked for the free card or put a 50% bet in, which would give us an easier time to get away from the hand if re-raised.
River is the A♠, meaning we've completely whiffed. Pot is $17.4 or $17.7, you have $11.20 in your stack. I assume the other player hand more starting the hand(always helpful to include that information too btw). I think any good river 3 barrel bluff is always dependent on the context of the board, how the hand played out and the player you're against. Given how you described the hand the villain seems to like to call when they hit. If we look at the board as a whole, T♦9♥Q♦T♠A♠, ask yourself your bluff, what are you trying to get to fold that could have called you all the way to the river, are they good enough to even fold when making it to the river? Most of what could have called you to river would be a pretty good hand that would absolutely call you on the river. You're essentially only beating a bluff at this point with your own bluff in my opinion. Anything in the A to 10 combo range either makes an easy call or makes a crying call, that given how they played the hand I believe they'd make anyway.
Personally given how the hand played, the BB seemed like a calling station, and calling stations aren't good ones to bluff with air on the river. In the end they show A♦9♣, which kind of confirms the calling station assumption as this is a pretty loose call to make on that river even with the 2 pair (mind you we're assuming you haven't gotten out of line to warrant a loose call from the villain). It's loose for the same reasons I mentioned for what you're bluff is trying to achieve, like what is A♦9♣ beating on that board, not a lot, even less that preflop raises and then 3 barrels it. Essentially you got unlucky that they called with something like they hand, however you also put yourself into a tricky situation to shove with air.
So the good in my opinion:
- Pre-flop raise late position is good
- Recognising this flop is good for our hand and deciding to build a pot
- I like the willingness to make a 3 barrel bluff
Things that are ok but maybe review:
- Your bet sizing is very big, you don't need to pot it everytime and you put yourself in a situation where you needed to shove to take a stab at the pot. For example, $2.1 on the flop, betting like 60% is like 1.20 with a call gives $4.5. A turn bet of something similar is about $2.7, so on the river you're looking at a $9.90 pot, with your stack of $15.10. With that stack you can take a much cheaper stab and fold to a re-raise easily. Basically you could have saved yourself money taking the line you did but betting better amounts.
Things that you could improve on:
- I'd recommend slowing down a bit and asking yourself what your opponent hand range may be, asking what your bets are trying to achieve, and can those bets make their range actually do what you hope it will do.
- Try understand your opponent better before making 3 barrel bluffs
Lastly just to answer your questions quick fire:
- Why did he call the first two barrels with bottom pair? Many reasons, maybe you gave them a reason to doubt you, maybe they're a calling station, maybe they're just having fun and don't care about the stakes, maybe they're new, maybe they love the gamble, etc.
- Could I have played this better? Yes, pretty much every hand can always be played better, the fact you're asking this is great though. Poker is a game where you should always be critical of your plays and learning from them! Don't beat yourself up, it's a learning experience.
- Is it less likely that he would call twice in a higher stakes game live? Absolutely not, I've dealt some nosebleed stakes in my day, if someone wants to gamble, or makes an excuse to believe they're ahead they'll call. The plus side is that if they are a spewy type of player you'll get them next time.
- Should I generally abandon bluffing a third time if the ace shows up since most people will have it in their range? Absolutely not, it's a valuable tool to have in your poker arsenal, however it's a tool that is very contextually dependent and is something that you need to be comfortable with blowing up on you. Live by the sword, die by the sword type of thing. If that is something you're not comfortable with drop it until you are, otherwise as I mentioned above just try do some range analysis on the hand, understand who you're bluffing, etc and you'll see better results. I think a better point about that board is even a lone ace in your hand there is a bit of a hero call in my opinion against your betting pattern.
- Or should I chalk this up to variance since he would have likely folded anything less than two pair on the river? Everything on the table has variance involved, it is part of the game, the skill part comes from understanding the maths, the opponents and when and how to apply that pressure. Chalking this down to only variance means you miss a learning opportunity.
Hope that helps, and of course this is just my own analysis and my own opinions so you'll likely get some differences from others, but it's always good to get an insight into others thought process.