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I don't think it would be inappropriate if you turn it the right way, asking if he would be bothered by making you a reference and stating that you're still thinking about their offer.
In your situation, I'd send an e-mail like this:
Hi [...]
Thank you for the generous offer, but I'm not sure I'll take it: I'm thinking of diversifying my experience ...
1
It really depends on your specific situation. Some of the things you could do include:
a) Try to get more involved in the work you are interested in. That means talking with coworkers on what they are doing and maybe coming up with ideas, improvements, etc. that you come up with.
b) Talk with your direct manager. This should probably be your first ...
1
If you have a good relationship with your old boss as it seems you do, you could tell him exactly that.
You love the company, the team, your former (and potentially, future) colleagues, the work they (and potentially, you would) do and you are very interested in working with them in the future.
You don't need to tell him about exploring other options, but ...
1
There is no exhaustive list when trying to learn what a company does, but I've found below helped me in my search
Every large company has a unique culture, and they normally put information about that on their hiring website - you can use it to learn about the company.
Most largish companies tend to use a mix of approaches based on different teams ...
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