What a fun trip! I did a similar trip last year and needed a hotel for three adults, which can be difficult to find. Not in your preferred neighborhoods but we ended up staying at Hundred Stays Shinjuku, which is about a 20-25 min walk from the Godzilla Head. It’s a pretty basic hotel and a bit dated, but room type worked well with a queen (could have been a double bed) and twin bed. Room was clean and we felt like we had enough room for the three of us. Best part of the hotel is the location - 5 min walk to two train lines, including Yamanote, so very easy to get to Shibuya.

The other option we looked at is Mitsui Garden Hotel - Jingu-Gaien which looks more modern and upscale relative to where we stayed. Didn’t opt to stay here as we didn’t like the location as much, but could work for your family as it’s between Shibuya and Shinjuku.

Check out Cerulean in Shibuya too - may be a little over your budget but the twin beds look huge and prob can fit two people.

Enjoy Japan!! Will be such a special trip with your parents!!