Pick one or the other and you'll be much better off. You'll make better progress and, in turn, enjoy your workouts more.
I tried for a good long while to combine barbells and bodyweight stuff, and perpetually sucked at both (some people may disagree, but that's only because they don't realize what's possible). Then I decided to get my act together and go purely with the bodyweight stuff, and less than a year later I can do a strict handstand press, at 6'1 210lbs (starting off with no handstand whatsoever). I wouldn't have ever got that if I continued doing both.
And even still, I'd say my training since August has been a wash. I busted my elbow up back them, which held me back, and this semester of classes had just been awful. Yet, I made better progress anything I tried with combination barbell/bodyweight work.
The idea of a bodyweight/barbell combination sounds good on paper, but it doesn't work in practice.
And yes, I'm actually holding that handstand. Get on my level.