Welcome to the forum! The Discovery 5 seems like quite a large vehicle for Japan - do you get any opportunities to drive it off-road there?
Here are my thoughts:
1. Tire size. Check this site, you can compare sizes available in your area with the OEM size:
https://tiresize.com/calculator/
In the US there's a sticker inside the door that lists the available OEM tire sizes. If you buy new wheels, you'll want to buy the OEM 19" size if you swap wheels and tires. There's a convenience factor in being able to swap on your snows yourself only when you need to, especially if you're only going to be in winter conditions occasionally. That said, winter tires are better in all conditions if you are driving in temps that are regularly below 45 deg F / 7 deg C in winter - the compound maintains its flexibility at lower temps, whereas summer or all-seasons get hard at low temps. I am running Hankook I*Pike RW11 tires in 275/55-20 on mine. I would have preferred Nokians but they sell out so fast here, and Bridgestones are overpriced, in my opinion. That said, the Hankooks are good enough, but far from the best winter tire I've ever owned.
2. TPMS. If you buy new wheels you absolutely will need TPMS or you will get a continuous warning at every startup that you will have to dismiss. In the US at least, any tire seller can install them, and they all work with the vehicle's receivers as they're an international standard, I believe. I imagine like everywhere, your LR dealer will charge a premium if you have them do it. However, my local dealer will swap wheels and tires for free in fall/spring and store your spare set for you if you buy the tires from them. Generally when you swap rubber only, all they replace is the valve core, not the full TPMS transmitter on the rim, which should last for several years.
3. Wheel size. Unless you already have a set of 19" wheels that you know will fit, there is little benefit to handling from going to 19" just for winter tires. If you will be driving in deep snow on unplowed roads, then a 19" tire will allow you to run lower pressures for extra traction, but these days tire compounds and tread designs are so good it's really not necessary to air down unless you're truly stuck. And the D5's Grass/Gravel/Snow program plus Low Traction Launch is superb.
Best of luck - post pictures of your D5 in your magnificent country when you get a chance!