and the place where I am going is in dative […] there is no movement involved
That's two times wrong logic. First, it's not about movement or not. It's about location vs direction. Let me show you with a different example:
Auf dem Marktplatz zeigt ein Wegweiser auf den Brunnen.
Nothing moves it this example. The market square, the signpost, and the standpost are all fixed in place. Yet, we have both a location and a direction. Thus, one auf takes dative, and the other auf takes accusative. (Auf is one of the nine two-way prepositions that function that way.) You could also rephrase as
Auf dem Marktplatz zeigt ein Wegweiser zum Brunnen.
That piece zum Brunnen is still a direction. As the preposition zu always marks a direction though it always takes dative.
Second, you have to remember for each verb whether it takes a location, a direction, or either. There is no way to tell what einladen wants from looking at the verb gehen for example.
The verb einladen can take both, but the place where you want to go to is a direction. While the place where you offer the invitation is a location.
And that's why it must be
Ich lade sie in mein Lieblingsrestaurant ein.
with in+‹Akk›, because you most likely mean that you invite her to your favourite restaurant rather than that you invite her while you are at your favourite restaurant.
Again, you can also have both:
Ich lade sie in meinem Heimatort in mein Lieblingsrestaurant ein.
That means, as soon as you and her are in your hometown, you invite her to your favourite restaurant.