period.

 
I will pay, and have paid, for lunch at the initial (non-BCD) M&G.  

 
I like going out to eat.  I like company while I eat.  And, the ceremonies of sitting, reviewing the menu, ordering, receiving the food, eating, paying the bill, helps fill some of the space if the conversation's not flowing.  If nothing else, we can talk about what one another ordered.

 
Besides, it gives me an opportunity to demonstrate, in a relatively inexpensive fashion, that I am generous (regardless of whether I truly am or not).

Is fine.  But paying her to join you is a no no

I agree that paying her to join you is a no-no.  But I do think it's reasonable to cover her actual expenses if she incurs them in order to meet you.  So a few bucks for gas, actual Uber cost, some babysitting $, are all reasonable asks.  But NEVER in advance via payment apps!  Only cash and in person!

I can see how when I said that I pay for a lunch M&G, it might have been construed as me stating that I'd pay the POT to join me.

 
Not the case.  I would invite them, pay for lunch, and maybe pay travel expenses (parking, etc.).  That's it.