If you have Composite, why not using it ? It is definitely a good product, and being node based like nuke and fusion, once you know one you can easily switch to the other two. Like Nuke, Composite is more towards high end film compositing, and not towards mograph. For pure mograph, afterfx is really better, but being layer based, it’s a nightmare to use for real compositing.
Also if you work professionnally like you say, you have composite licence with max. Nuke is another max/composite seat price, nukex around two of them. So, although you are sure to have the budget, I wouldn’t learn a software that I won’t buy anyway. Of course, once you know Nuke, you pretty know Composite too and vice versa. In a week or two, the other will be familiar.