Is the basement heated? We have Nu-Heat under the tile in our M-bath. You can go to their site and figure cost per square foot for the product. It's easy to install. Your tile person can lay it under the tile -- easy to hook up the thermostat, but you'll need a designated elec. line. We also have hot water tubing under North-facing rooms in our house -- a retro-fit thanks to a stupid HVAC guy who didn't plan adequate forced air heat for those rooms. Had we known we needed the hot water heat we could have adapted a water heater for supply; we had to buy a separate tiny boiler. I wish we had the hot water heat under ALL our flooring. Those are now the most comfy rooms in the house!

I tried to answer your question in the kitchen forum but it said I was spamming. If you google warmly yours they have an online calculator. It worked well for us. I am retired & not at all connected to any company.

Your HVAC expert who is designing the system should explain the requirements that the manufacturer has for their particular system. In general, in your situation, you will need lower temperature loops closer together rather than the traditional higher temperature loops further apart. The whole system has to be designed for the water temperature that you will need, as well as the PEX placement distance. Floor coverings also has to be taken into account in the plan. Rubber and carpet actually have insulative power (R values), and are not conductive to easily passing the heat through into the home. That has to be calculated when designing the system, and it will increase the lag time to temperature changes, which is already radiant's biggest drawback.

The best approach for both locations is the one that your hvac and general contractors are most familiar with.