I think from their initial email I would have had the same reservevations you had.
I had a similar offer not from a publisher but from a self publishing writing couple, it was a 70 page childrens book and they wanted a price per page, I told them that doesn't really make sense as it totally leaves out the development process.
I hit them with a range of questions after their initial request for a quote (oh and the y didn't share the manuscript so they were asking me to pull prices out of the air)
here's a section from my email response
** You mentioned it's 60 - 70 pages, are these all illustrated?
Do you know how many are full page, 1/4 page, 1/2 page illustrations?
Will all illustration require background or will some be spot illustrations i.e floating no background?
The images that you attached could you describe what elements you like about them, this will be easier for me to produce illustrations in my style that you will like?
Do you have a budget and timeline in mind for this project?
What is the physical size/dimension of the book being produced.
It is hard for me to give you a price per page as there would be a number of factors I would have to consider, for instance one being the agreed turnaround time, another being the agreed rendering of the illustration and complexity of them. **
Maybe some of the above questions may help you, they also wanted me to sign over copyright of images. So maybe you could ask what will the usage rights be, are there royalties? Also I would suggest doing a sample spread but I recommend agreeing a price for the sample first before putting a pencil to paper. I did a sample for that Job and was careless and trusting, I managed to squeeze out $50 after which was laughable.
This project didn't go ahead and I'm delighted it didn't but I wish you the best of luck with this one.
Try get as much info as you can, and remember you can always give them an estimate and then if terms change later before you sign a contract, you can explain this and re evaluate your estimate.
Hope this helps
Best of luck