That’s a great question—and you’re absolutely right about the general rule.

Grammatically speaking, invitar is a transitive verb, so when it refers to a male person as a direct object, the expected form is indeed invitarlo. The use of the personal a doesn’t change that, as many verbs that take direct objects also require the personal a, just as you mentioned.

So why does invitarle appear in the lesson?
What you’re seeing here is an example of leísmo, which is very common in Spain and is considered standard and acceptable in that variety of Spanish when referring to male persons. In those cases, speakers often use le instead of lo, even though the function is still that of a direct object.
To summarize:

Invitarlo → grammatically correct everywhere, and the safest option across all Spanish-speaking regions.
Invitarle → reflects standard peninsular Spanish usage (leísmo) and is also correct in that context.

Fluenz exposes learners to these real-world variations so you can recognize them when you encounter Spanish from different regions, even though we generally reinforce the traditional direct/indirect object distinction as the core rule.

You’re thinking about this exactly the right way—nothing is “wrong” with your understanding. It’s just one of those cases where usage and regional norms add an extra layer on top of the grammar rules.
Best Regards!