First, you can definitely include it at the end of your workouts, maybe unless you totally exhausted your shoulders (push days) or are just too exhausted in general. I would suggest this over adding handstands at the beginning of the workout in order to have more energy for the main workout and also be warmed up for the handstands.
Second, unless you are practicing for a long time or are doing something like handstand push-ups, there is no reason not to do it on non-training days. Practicing a few handstands will not be so hard on your muscles that it would hinder recovery substantially (unless you are a pro athlete maybe, where every last bit of recovery may count). I didn't totally understand your training schedule, but I guess it's one day push, one day pull, one day leg (Mon: Push OR Pull / Thu: Legs / Fri: Push OR Pull)? Then you can definitely do it after pull/leg days, and as I said unless you're overdoing it even moderately after push days.
As a side note, if by "I can’t train 1-2 days after my workout" you mean that you are so exhausted you literally can't do any physical activity, you'd be overdoing it. Otherwise, yes you are right you need that recovery time, but as I said a few handstands are still fine, you don't need to basically lie still all day to recover from a workout.