Its not very difficult to travel between heathrow and gatwick.
There are a number of options to get from LGW to LHR.
The simplest is by Bus/coach there are several companies that do that - are very frequent though can be expensive around £18 if I recall. The same companies also do other routes i.e. to STN. You are also subject to Traffic on the M25 (the main ring road around the city) . Journey times can be as little as 35/40min to as long as 2 hours in extreme traffic conditions, average journey time would be around 1 hour plus the time from the terminal to the station/check in desk etc.
You could also get the trains and go via London Victoria and then Tube out to heathow but invariably it would take you just as long. Also it is an awkward way of doing it if you have baggage.
BB is on the 75/76 fleet as he has longhaul rostered.
LGW has both 737 and 757 based although which type of 737 I'm not sure.
Someone else might be able to shed more light but I believe LGW is a very popular base with the 737 guys so it may be difficult to get into first - but I'm by no means sure on that.