Now we are asking the right questions. Unfortunately there is no easy way to search for that. Some hotel sites do have マット as a search option but it's really not very helpful.
That's because quite often the hotel would have only one room with big enough bathroom to have a proper mattress. Also it seems that even if they had them before when they break down the hotels are not replacing them.
For what it's worth B-Side in Gotanda at least used to have mattresses in some of their bigger rooms. Also Rochelle in Kamata has some rooms with bigger bathrooms than an average Tokyo apartment and some of them had big mattresses though some of them were replaced with thin versions last time I visited.
The only place with mattresses in Yokohama I know is near Motomachi-Chukagai station and it was maybe Akai Kutsu though I am not sure and it was already several years ago. Though the place was pretty great, big fat mattresses and economical long time stay prices.
Edit: was on my phone and somehow managed a dozen typos and even fucked up both the name and the location of a hotel: it's B-Side and in Gotanda. And I have probably been there more times than in any other hotel in the Greater Kanto area...
https://rochelle.hard-love.me/
https://hotel-b-side.com/
http://www.hotel-akaikutsu.com/