You can get by with a cheap Class D amplifier without sub-out and still run a sub so long as the subwoofer has high level input/outputs. I ran both a Lepai LP-2020A+ and a SMSL SA50 (both 2 channel amplifiers) and a powered subwoofer for a while.
with your budget look at a SMSL SA36/SA50/AD18, a pair of Micca MB42 speakers, and a Polk Audio PSW10 or Acoustic Audio PSW-8 sub. It should be obtainable for that budget if you just convert AUD to USD, no idea on how things are really priced on your side of the pond.
smarter route (but over budget) to go would be to go for the AD18 or a Klipsch PowerGate/full size receiver along with a pair of Micca RB42 and hold off on the sub. Save up a bit more money and add in a subwoofer later.
RB42 are less efficient than the MB42 and need more power but they have significantly more low end than the MB42. I wouldn't pair the RB42 with anything lesser than the SMSL SA50 and even that wasn't enough for me, I made the move to a PowerGate.