Learn English – {pay / give} compliment

There are two possible contexts...

1 - you're taking him all the way back home with you,
or
2 - you'll be dropping him off somewhere else before you get home.

I'll pick him up on the way home is far more likely. This version works for both the above.

I'll take him on the way home only works for the second meaning. This is because on the way home means during the journey home (you can't take him all the way home until the journey is finished).

It might help to consider these two related constructions...

3: She asked him the way home
4: She asked him on the way home

...where we don't know what she asked him in #2 - all we know is that she asked during the journey.

TL;DR: Both are valid, but #2 has a very specific meaning that usually isn't likely to be intended.

The way I would most likely use time-slots is as follows:

... Could you tell me which time-slots are available?