I will simply during the timeout for injury approach the red flag and politely tell him that while the station may be going to a commercial break I will not allow him to interrupt the game and to please go to the sidelines and we will return to play as soon as the injury timeout is over applying the proper timing mechanics. Thanks, Guys.
Can not address televised NFHS contests, but I can say with authority (14 seasons of FBS football - 8 as referee) that, in NCAA, we do not have authority to circumvent the TV process. Like it or not, when they want a break, they get it (even if it may be beyond their pre-planned set of breaks), and we don't resume play until the liaison ("red hat" - not "red flag") releases us. I can just imagine how quickly you would be fired if you consciously refused to go to break, or deliberately resumed play before being released by the liaison. Since they have contractual agreements with the various conferences, the TV folks would be all over the Commissioner. The Commissioner would be all over the Coordinator. And the Coordinator would not need to be all over you - he'd just fire you immediately, and have no need to talk to you, ever again. (What is it that flows downhill?)
Having said that, when talking to TV producers pre-game, they almost always made the claim that they wouldn't go to break following a "sudden change," like an interception, so it wouldn't disturb the natural momentum ebb and flow. And they almost always kept their word. But, in one game I had, the liaison stepped out on to the field with his arms crossed (universal signal for a break request) after a big interception. I just looked at him, shook my head, and pointed to the ground, i.e., "stay here," and a moment later he backed off and we played on. No repercussion, as it was the right call, per TV's own process. Now, had the liaison stayed out and insisted on the break, I would have let them have it, and I would have put that in my game report.
As for game clock / play clock status when returning from a TV break, it was simple for us. For the purposes of clock status, the TV break never happened. They both simply obeyed the rules for whatever caused the clock to stop to begin with. After signaling and announcing the T/O, I'd always immediately check with my S to confirm the clock status when we come out of the break. Likewise, I'd confirm the play clock status with the B, and make sure it was set for the correct duration for when we resume play. Then, after the break, like the really good S he was, I'd look to my S and he'd always be echoing that game clock status as I was about to signal the ball ready for play. Never had an error or issue.