I would definitely not do this on a Saturday. Get up early or get the car the night before. Skip this festival; it's probably lame and a tourist trip. Or you could swing by if you have time.

If you need to park the car overnight and your hotel is not parking friendly then coin parks often (well it's a free market) have low rates overnight like 500 yen or you could find a meter that is not restricted.

Take the Tomei to the south side and take the non-toll road to 2500 m. Or you can also go to the Gotemba trail head, which is at lower elevation. This is literally only a 5 min detour from the main road from Gotemba toward Fujinomiya. Here the ascent would start with massive lava fields, which is pretty cool.

p.s. You are aware that you need to get a silly intl driver's license from AAA?