I don't think there is anything wrong with inviting a child along to keep your only child company (or indeed your not an only but significant age gap child - DD had a friend 14 years younger than her siblings who did this).
However there is something wrong with expecting them to pay when you honestly are inviting them for (completely understandable) selfish reasons, and when you take it as a given the child will be made available to keep yours company.
I am very glad our almost neighborly finally moved, as it got to the point where we had to be blunt to the point of almost being rude about our ds1 not being some kind of companion for hire. Initially they'd invite him to free activities, then there was something paid which I sent money for, but it escalated fast til they were going to places like Legoland and very expensive water parks almost weekly, and we started saying no - the trips were too frequent and we said no a few times due to clashes with our son's sport fixtures, and got "well can't he miss a match every now and then?" He didn't want to miss matches, he has a strong loyalty to his team and is the kind of kid you have to lay down the law to to stop him playing with a broken leg
They tried to reschedule to "whenever DS has time" at which point we said it was getting too expensive which was unfair to his siblings, and then they offered to pay. DS revealed that didn't even want to go any more as he always had to do what their DS wanted because he was invited to keep him company. It seemed likely by that point that if he was being paid for this would get even worse!
The other child's mother became quite unpleasant about the withdrawal of DS's services as companion, especially after inviting another friend didn't go well because he was less complaint/ well behaved, and she came back to us, telling me i had it easy with 3 kids because they could entertain each other and it wasn't very nice not to lend her one as i had a spare! That could read as a joke but it really didn't sound like one when she said it... She spent a lot of time complaining how hard it was to entertain her son, how demanding he was without our DS to entertain him, and how she and her DH got no peace, yet also started making a lot of digs about it not being possible to really give 3 enough attention and that they must miss out on things because 3 must be too expensive...
Once it became apparent she viewed our DS as some kind of poor relation type companion I felt pretty pissed off we'd coughed up for so many trips purely for her benefit, and had to be quite hard about refusing further invitations on DS's behalf even though she then when in danger of losing his service, offered to pay.
DD's friend's parents never put the same sort of pressure on and just invited someone else if DD said no - which of course is all great and as it should be.
The mother in the op keeping on coming back with offers to pay half only after he said no, and then in full when that was a no, reminds me of DS's friend who became more of a frenemy.