Have been working on a US domestic trip, but probably applies for international as well.
Have used booking.com for research. In those instances where they had better rate than the lodging site, I contacted the hotel and explained what I found. In both cases they bettered the deal.
Once when I encountered such a situation the hotel suggested I book the room through booking.com, which I did, saved some money and got the same thing.
Will give the hotel a chance to eliminate the middleman, but will choose the lesser cost when it is equal.
Just proves there are no absolutes, but "if you don't ask, you don't get"
Happy affordable travel to all.