I think he'd do fine. He'd probably just be a little more serious.

He's already a perfect fit. I'd see him becoming a bit more gritty with all the weight that would be on his shoulders.

He'd act the same, the only difference would be is that he wouldn't have this grumpy old man yelling in his ear telling him how to dangle a villain over the side of a building :P
 
But no, I think we got a sense of how trained Terry was in the last episode of JLU which spotlighted on Terry and was the "official" final episode to the Batman Beyond series, even though Bruce wasn't dead, he was getting there though, but Terry did everything, from breaking into Amanda Waller's mansion, to working with the Justice League to help them take down most of his rouges gallery, which he did all on his own without Bruce's help or advice. I think he would act much the same as he does now, maybe a little darker/serious yes, but the thing that makes Terry a great Batman is that he does things differently than Bruce, like speaking to villains and taunting them (Spidey style), for example in the movie Return of the Joker, he beat the Joker by messing with him, like Bruce, Terry didn't find any of his jokes funny, but unlike Bruce, Terry actually taunted the Joker possessed Tim Drake by making fun of his jokes and laughing at the Joker because of them, something Bruce never does with villains (like laughing, or even smiling for that matter), which in the end gave him the edge he needed to defeat the Joker and have Tim.

" He's already a perfect fit. I'd see him becoming a bit more gritty with all the weight that would be on his shoulders. "

  In Mainstream U, isn't Damian the one teaching Terry?  Not Bruce.

As we saw in the Epilogue episode, when Bruce dies, Terry will be mature enough to handle being Batman without supervisor.