Pets adoption pool change

I am doing the Nature's Calling event and got to where I have to adopt a dog. I already have one female dog in my family and wanted to use this opportunity to give her a partner for breeding. But alas, all the adult male dogs in the selection are neutered and therefore unusable for me. What can I do to get new selection? Is there a way to affect that (including by mods like MCCC or UI Cheats)? Is there a cheat? Or do I simply have to wait? And how long until different pool will appear? Surely appreciate help with this.

And, if I decide to adopt neutered dog to have this part of the event done, is there a possibility to remove neutered state from a dog? Is it a trait or something I can remove with UI cheats? I never played with neutered dog, so I have no clue how that work.