You don't say what type your girls are.
I ran my Shetland tup with my flock year round and didn't have any problems. Although in theory you can get lambing going on for ever, it's not too difficult to keep an eye on just two! Plus, in practise, less than two cycles is usually plenty, unless you have some early-cycling (like Charollais) and others later (most primitives), or there's a fertility problem. (Thankfully rare.) It starts to be more of an issue keeping the tup with the flock after the first year, if you have retained ewe lambs that you don't want tupped.
I have used a Shetland tup lamb on a mixed group of girls, up to and including a North Country Mule. Rosie Mule was a stretch for him, but as long as you have some uneven ground, or a slope, love will find a way
You could also try finding someone who wouldn't mind your girls coming to visit their tup while he's working; that's how I got my first 4 girls tupped.
As others have suggested, buying a tup lamb and then eating him is a tried and tested technique.