Well, going from very poor grass to very rich grazing will probably make them scour. I wouldn't worry about putting Primitives on there, but I don't know enough about Suffolks to know how well they naturally restrict their own intake.
You could use an electric wire to restrict them to grazing strips to be sure they don't get bloat, staggers or whatever commercial type sheep get from too much green grass. It would be cheaper than getting it topped if you can't do that yourself.
We have just cut a pasture which sounds exactly like that for hay and in spite of cow parsley which has gone to seed and the thatch of the earlier hay, the crop is looking fair. It is currently getting utterly drowned in a daylong downpour, but we have several days of breeze and sunshine to come, so I think we'll get something 'to fill their bellies' as my neighbour puts it ie not good hay but we're no longer breeding anyway so quality is not so important. Perhaps making hay is an option for you?