What should I use to smooth out and finish gap in trim for deadbolt lock latch?

I’m installing a deadbolt lock that swings across and the latch has to be flush to the door. This trim is extra wide so I had to cut into it so the lock will catch the latch. I’ve purchased epoxy wood putty I was going to smooth it out with and paint. Is this the best solution or what’s best to smooth out the edges and inside parts before paint? There’s wood and drywall in the hole.

Here’s pictures of the parts: (https://i.stack.imgur.com/N9qOU.jpg)

Here’s the latch in the space: (https://i.stack.imgur.com/72ABk.jpg)

Here’s what it will look like once installed: (https://i.stack.imgur.com/7Y3N0.jpg)

Just need to make the pieces next to the latch look nice then paint before attaching latch

The reason I’m using this lock is for a pool enclosure, it must be a bolt or chain type, 6 ft high. The door has to be safe from kids exiting the house toward the pool, not for home security.

This specific lock is smart home compatible so I can have it auto lock the latch when the door is locked and auto unlocked when a keypad is pressed on the outside it will unlock both locks.

I can install it I just want to finish the space inside the hole so all the wood and drywall isn’t so rough inside the latch part.