I have been flying the 38 alot as of late and the biggest tip I can give you is ...
Manual Trim only ... DO NOT USE Combat Trim. If you use combat trim, your success in the 38 will be abysmal to say the least.
(Me in a 38, enemy in N1k) As I close on their six, at around D600 they start a hard climb up and to the right.
You are coming in fast in the 38. With the centernose guns, you should already be firing a guns solution at that distance with the closeur rate that you have. Anyways ...
I'm coming in slightly faster so it takes me a little bit longer to adjust and get guns back on target. As I'm doing this the plane starts to roll inverted and goes back to the left.
Decision time ... Too fast ? Can I still track and not lose speed ?
At this point I usually chop throttle and continue to track in order to prevent overshoot.
Here is where you went wrong IMHO. At the intial engagement, I would have aready chopped throttle. At this point, I would have firewalled the 38, trimmed nose up and gone vertical.
During that barrellroll, I would bet that he has lost sight of you and is praying for the overshoot right in front of him. You had more E than he did at the start, so without making a real drastic move for firing, you should still have more E that him, especially if he has barrellrolled.
However now they are higher than me and I've lost E. If I don't chop throttle I overshoot right away, and now they are on my six with a pretty good shot.
Like I said ... you really should be above him at this point and watching him. If he is slow, which I presume he would be, I would start trimming nose down and beginning a roll over for another attack.
Occasionally (when I chop throttle) I'll get one decent shot on them before wallowing around and getting minced. Once I get slower there is no recovery due to the better acceleration of the N1k.
This could be because you have combat trim on. If you have the slightest bit of and E advantage, and trim out the 38, the N1K should not catch you.