Whilst the airlines may not care what FNPT II (MCC) device was used to get your MCC certificate, you might!
A jet FNPT II (sim) will stretch you more than a turbo prop will and so will test your capacity more. That stretching of your capacity is part of what the MCC is about - forcing you to prioritise your task load and operate in accordance with the SOP to get the job done under demanding circumstances.
If you then do a selection ride, the extra speed of the jet will help - if the sim ride is done on a jet sim then you are already up to speed and if the sim ride is done on a turboprop you have capacity to spare (as a friend of mine has just found out on his Flybe selection).
As for 737 vs Hawker800 vs any other jet, what matters is how it is operated rather than what it is. ATP at Southampton used to have a Caravelle (how many of those do you see these days?) but it did not matter. Flown to an airline SOP and the appropriate flight profiles the end result is the same.
BA, MYTravel, Britannia and others for whom we do training that goes beyond the basic MCC require a jet - and they don't really care what sort. The handling and operating skills that the extended courses (Jet Orientation Course etc.) give the trainees really does smooth their path on the type rating. TJV is spot on with the MYTravel feedback, by the way.
In my opinion - don't worry about whether or not it is a 737 or Hawker, as long as it is a jet FNPT II then you will be better placed. More important than that is to try to do some extra above and beyond the MCC - which can do little more than give you the very briefest into to airline flying. Use that extra FNPT II (MCC) time to get your multi crew skills up to a level where they are second nature as THIS, more than anything else you do on the whole ATPL course, will ease your transition to A320/B737/ATR42/Dash 8 etc.
WWW makes some bvery good points about the other facilities - and remember, there is no law that says you have to do groundschool, flying and MCC at the same place.