NEED SPECIAL DOORS - CAN ANYONE HELP

I want to source doors with a bottom panel that is mesh, with a solid panel that can be pulled down for privacy. I do NOT want a screen door per se, it is for use as internal doors in our home. I’ll have to go into a long winded explanation, so bear with me. But I know I have seen images of what I want on the internet.

We have a small cottage and an efficient wood stove (its cold down here in Tassie), but when the bedroom doors are closed, all that heat is trapped in the living room. We have 5 inside/enclosed cats, and 3 don’t get on with the other 2, so we practice separation. However, separation involves closing bedroom doors for long periods, resulting in cold rooms at bedtime.

I’d like to get doors with the lower panel or panels (I’m thinking of usual panelled internal doors here) being a slide down or hinged arrangement with strong cat netting in the space. Thus we could have warm air flowing into the rooms but the cats could not physically contact each other, and we can shut the panels for privacy if required.

If anyone at all can help me I'll be most grateful. I don't want to spend a fortune, and we only need 3 doors. In Tasmania would be great but I'll consider interstate if freight's not going to be prohibitive.

Not necessarily mesh, but a door with a grille, as seen commonly in offices may be an option.

However, if I understand it correctly, heat rises, so that plan may not work.

A common way to achieve is to install a heat transfer kit, with a ceiling grille and duct with an inline fan. These are available with one or more ceiling outlets.
Could be done by handyman or DIY, except for the electrical Connection for the fan.