The friendly bus does go from HRT to Haneda, but the only caveat is the first one from the hotel is at 9:10 am. The more frequent and earlier ones leave from Shinjuku, but by the time you get over there, you might as well take the trains.
They do also sell out, so I’d make a reservation. I got caught out with multiple bags and ended up taking an expensive taxi to HND last week. Think I’ll probably leave the steamer trunk at home next time and ship my souvenirs instead.
I paid 17k ¥ for HRT but that was a Covid booking. It was 30k as I was leaving.