Hi there, career Dragoon here.

You can go for about 4:20-4:40 while sticking the rotation down to the letter and managing your Invigorate pretty much perfectly before you run out of TP. At that point there's usually about 20 seconds (or so; can't remember) until Invigorate comes off of CD. Without a Paeon pre-empting that, your DPS will go down the toilet. Monks and Ninjas will have their own exhaust times, but will still hit that wall.

Honestly, I think a Bard is pretty much a mandatory spot in a Raid. I get it if people don't want to play it, but they bring incredible utility and can still do decent DPS. Plus, the fact that they can attack whilst moving from a range and have no positionals is often a great boon for dealing with mechanics (T8's Brainjack being a good example).

TP is finite resource, and it will run out without a Bard. Simply putting delays between your abilities is no help at all - I imagine that the lower overall DPS that someone doing this would have is about even with the DPS of someone who pushes hard and then has to sit around like a lemon waiting for Invigorate to come off cooldown.