I think you're missing a key point about splatoon and a lot of competitive objective based games.
And that it's the team with the best push that wins, not the team that possesses control the longest. For example, you could get the rainmaker to 20 points, keep control for 3 minutes, and still lose. Why? Because without a knockout, your team has to defend and try not to lose. It's not just "Hey we got a good enough lead, let's squid party", trying not to lose is serious business.
Luckily, for you, trying not to lose is way easier than trying to win and make a comeback. Just because the enemy team needs to move the objective further than your team initially did, and they have to regain control after you pushed. So there's something good there. So what to do after that? How do you make sure you keep that condition going, because the enemies are trying everything to stop you.
Well it's exactly that. The enemies are going for risky plays, so be sure to punish them once they go out of position or try to make them run away with your sub or something. Even if you trade, the game is still in your favor because you don't have to do anything extra to win, you just need to make sure your lead is in check. But assuming that you are keeping control until overtime (which is what I got out of this post anyway) you're doing this right.
So for stopping pushes in overtime, some key things to have on you.
Map control is #1. This dictates if the enemy team even get to move forward most of the time, so it's rather beneficial if you ink the ground to prevent this. (Plus you get more specials by inking more ground, so that's cool)
Your special is #2. Specials can start pushes, but they can also end them pretty quickly. Every special in the game can start a push and end a push (obviously some are better than others at this), but specials can be used to stop or start momentum for your team.
The best defensive tower control special is the Stingray (by a longshot). When used from spawn, there's literally nothing the other team can do to counter the stingray, they either get off the tower, or die and get off the tower.
However, like you said, you don't main a stingray weapon. You play the dualie squelchers with tenta missles. Which aren't the best for clearing the tower, but we'll work with it.
The tentamissles (as you probably know) lock onto multiple targets, and then send an air strike of missiles raining down upon the enemies. The missiles lock on to wherever the enemy previously inhabited. This is pretty good for clearing the tower for 2 reasons. 1 It forces the opponents to move (somewhat like stingray) or die, and 2 it gets stronger whenever the opponents are huddled together (which happens a lot in overtime). Allowing you and your team some time to tap the tower, while the enemy runs away. Even if only 1 person is on the tower, and you lock onto them, guess what. It's only 1 person left on the tower, get rid of them and touch the tower real quick to secure a win.
This applies to other modes as well: bomb rush / bubble blower in splat zones, stingray / inkstorm in rainmaker, and bubble blower for clam blitz. These are just the specials that can reclaim the objective and control of the map quickly to help you stop a last second push.
Last tip I have to avoid losing in overtime is to maintain your numbers against the enemy team. As the enemy can only start a push in overtime if they have a numbers advantage against your team. Having a 4v4 matchup isn't enough to let the enemy roam free on the objective. Even 3v4, it's hard for the enemy to comeback from in the middle of overtime. The only time I would start worrying about losing in overtime is when only 2 or less people are alive on your team. Because that's literally half (if not more) of your team, dead. Making it pretty easy for the enemies to start a push and end the game with a win on their screens.
