You are right, you can do it without a solicitor, if you are happy with the model contract you have found. We did get a solicitor's advice - but it was for free because she is a friend.
I am a lesbian mother with a 4 year old birth child. We used a known donor and he is an involved father - sees our daughter every week, pays child support, is named on the birth certificate etc. We used a known donor because we wanted our daughter to know her father: once we had decided that, it made sense to kind of give him from the start what a court would be likely to give him if we did fall out: regular contact, consultation on major issues etc. In return, we expected him to make a financial contribution and leave us to get on with the day-to-day parenting.
In the first year, there were some prickly moments when the contract was useful: i was able to point to it and say, "This is what we agreed" and use that as our default position. As time went on, we have grown to trust each other more and be more flexible, but I think we're all glad to have a contract. Just drawing it up was a great help, forcing us to discuss the difficult issues at a time when all of us just wanted to be nice to each other.
I don't know if this is the kind of arrangement you have in mind, but whatever your level of involvement I think a contract is very useful - so long as you are both aware of what it can and can't do.
Have you made contact with the Donor Conception Network? i think they have lots of single members.
You may also find it worth posting this under 'Lesbian and Gay Parents'. I know you're not lesbian, but that's where you'll find the most expertise on this issue!
But I'm also happy to answer any further questions. It's a fascinating subject!