I think the previous answers give a lot of good answers. Personally, I would have lunch, and then pay. If the customer says "this is on me" or something similar, I would argue "No, you are our guest, let me get this", but if he argued that even once, I would give in. After all, the customer is always right (theoretically).

However, I would finish with the usual "ok, this time, but next time it is definitely on me".

Should I offer to pay the lunch instead?

Yes, in fact you should assume you are going to pay, and act like you expect to pay.

I have never been in situations like this and I am not sure what to do. 
I would not like to offend him by not paying the lunch 
(if it's expected), but neither to offend him by saying that 
we will pay the lunch what he had offered first. 

Some people would be offended if you did not let them. It is like a mark of authority, and I assume your customer has a bigger company than yours. So expect to pay, attempt to pay, but do not labour the point nor argue it.

I think the previous answers give a lot of good answers. Personally, I would have lunch, and then pay. If the customer says "this is on me" or something similar, I would argue "No, you are our guest, let me get this", but if he argued that even once, I would give in. After all, the customer is always right (theoretically).

However, I would finish with the usual "ok, this time, but next time it is definitely on me".

Should I offer to pay the lunch instead?

Yes, in fact you should assume you are going to pay, and act like you expect to pay.

I have never been in situations like this and I am not sure what to do. I would not like to offend him by not paying the lunch (if it's expected), but neither to offend him by saying that we will pay the lunch what he had offered first.

Some people would be offended if you did not let them. It is like a mark of authority, and I assume your customer has a bigger company than yours. So expect to pay, attempt to pay, but do not labour the point nor argue it.