Running suspension with narrow tyres is a fool's game: the store is trying to chump you. How far a tyre pressure decrease will take you depends on your weight.
To be honest, the BadBoy is just a design driven by marketing. It's aimed at people scared to ride drop bar bikes who still want to go fast and imagine thin tyres will help them do that...
In fact tyre width has nothing to do with tyre drag - wide tyres can be extremely fast (it depends on the materials used and other details of construction http://www.schwalbetires.com/tech_info/rolling_resistance) until you get up to speeds where minor differences in air resistance due to width matter - which you won't on a flat bar bike! These differences only come in to play at serious racing speeds. (The fastest wide tyres are probably Schwalbe Almotions; they cost several times more than the cheapest.)
Good suspension components are expensive and usually require a lot of maintenance, and cheap ones can be dangerous.
..If you're a light weight type of guy, try fitting the widest tyres you can and dropping the pressure. Otherwise, buy a bike with room for wide tyres. Probably avid suspension - good suspension is expensive and requires regular maintenance.
...A really good bike for this sort of job is an old 26" mtb with no suspension, bought off ebay - Kona and GT are good bets. Or you can buy a new boutique adventure bike that is a copy of one of these for about 5 times the price.
Also: if you're going to ride on gravel etc, get some tyres with some knobs for grip - they will cost you speed and handling on the road, but washing out on gravel can be painful and dangerous.