This idea might work for you, too:
When DS was in afternoon kindergarten, we had the party in the morning. Yes, all the kids were invited and they went straight from the party to kindergarten. I warned the kingergarten teacher; she appreciated the "heads up."

I didn't want anyone to feel left out either, especially since they were going straight to school. I wrote on the invitations that if anybody needed transportation or needed to drop their kid off at my house on their way to work, they could. Nobody took me up on it and every single kid came. Here's the best part:
it wasn't just a birthday party, it was really a field trip. I just called Peter Piper Pizza and asked if they were available for field trips. They were!

We were asked to come in before they opened. We got a great "backstage tour"

of the facilities. I was impressed; the tour was much longer and involved than I had expected. At the end, each kid made a personal sized pizza to eat for lunch. They were also given two tokens as part of the field trip and I purchased four more tokens for each kid. DS still talks about it. The tour was pure field trip. The employees knew it was his birthday, but the only special treatment he got was getting to be first in line. When everybody had their pizza, we sang "Happy Birthday" and blew out the candles on the cake. One very nice benefit to doing it this way was that our kids were the only ones in the restaurant. It was very easy to monitor them and the door was locked until they sat down to start eating. Perfect!
As far as no gifts, I completely understand. We had one of DD's parties at the Children's Museum. Each kid came with a wrapped book labeled for girl, boy, or either instead of a present for DD. When it was time for presents to be opened, the presents were distributed according to the labels. So each kid got to take home a book and this also served as their party favor instead of bag of junk.
It worked very well for all but a couple of the kids. I had some extra "either" books to trade in case somebody got a book they truly didn't like. Except for the 10-year-old boy who kept sobbing, "I don't
want a book! I want a toy!"

, it went pretty well. If I were to do it over again, I think I would schedule time for them to trade with each other after the books were opened and maybe open the presents one at a time.
You could modify the idea to be toys instead. I asked for books $5 or under. Some of the books looked like they were more than $5, but at least the parents knew that wasn't expected. As long as all the kids are receiving the same type of thing, it would probably work.