Can I use friction shifters with a clutch style rear derailleur?
This is a good question. I speculated about this over the weekend and came to two conclusions.


Friction shifters were designed for narrow(er) rear cassettes. The largest cluster I've ever seen a friction shifter work on is 8 speed. I don't think it will pull enough cable for you to access all the speeds in your ten speed cluster. EDIT: As an insiteful commenter noted, the cluster is only 1.8 mm wider for a 10 speed than an 8 speed so it should pull enough cable for your needs.
Trimming will be more difficult than with index shifting. The industry has added several gears and not much space to the rear triangle. The result is thinner gears and chains, so dialing in the rear trim will be more difficult. 


So, will it work? Probably. Is it ideal? Probably not, but it's worth continuing research about the topic. Just my 2 cents.