What is the easiest way I can send one contact's phone number to another on my Android?
Copy and paste from Contacts app to Messaging app should just work without any problem, I've used it a few times before to copy lines from addresses, emails and that sort of thing into an SMS.

Are you sure you long-pressed long enough in the SMS message to get the input menu up with the Paste option on it? Or that you copied it properly in the first place? Copying data from a Contact is most easily done by Editing the Contact, then long-pressing on the field you're interested until the input menu comes up with the Copy All option.