Does modeller provide a means of generating models that differ only
slightly from an input template?
Sure - you could build some models using a simple 1:1 alignment to the
template. The final models will look very similar to the input (by
construction) but you should expect to see some variability,
particularly in the regions that are not strongly restrained (e.g.
sidechains). You could add in some extra restraints around the docked
region, if you have some extra data or intuition about the interactions.
But Modeller is designed to generate good solutions to your scoring
function, not to sample from an ensemble - if you want to do that you
might have more luck with an MD package.