Greetings
I would say there's just no way. Remember you will have to clear immigrations, get your luggage, clear customs, then get to Gatwick and re-check your luggage and check in for your flight.
Security is very tight these days, London traffic is bad and Heathrow is an absolute, total nightmare when it comes to luggage. Right now, due to extra security, a backlog of delays due to bad weather and strain on a system that is handling 20% luggage than it's designed to, people are waiting 3+ hours to get luggage and there are 1000s of lost bags still being sorted out. I have a 100% record of lost luggage with BA transfers at Heathrow.
There's a chance you might make, but one security alert, long line, delayed flight or baggage snafu and you're vacation is at risk. Flights are very full these days so you could wait a long time for an extra seat. $150 may seem a lot now, but it's much cheaper than a completely re-arranged holiday or having to buy clothes whilst a suitcase is AWOL.