Unless you are looking to fly say a hundred plus hours in a very short period going to the US will not be cheaper when the cost of the flights to the US, accommodation etc etc is taken into account.
Getting a friend with an interest in riding along in the right hand seat and cost sharing would reduce the cost - so plan a very, very long cross country/tour around the states - but the US can be difficult in winter due to weather as well.
The current issues they are having with the shut down is impacting issue of airman certificates so the piggyback may take a while.
Overall flying in the UK may be cheaper but the weather will get you too.
If you can get the licence sorted I would suggest renting in Florida and flying in the Bahamas - the allow LSA now so a way to keep the cost down a little.
You biggest problem is going to be licences, you have left it a little late and US politics seem to have bitten you too.
I will suggest this option - Barbados - in the winter the weather is pretty much perfect - there is a Tomahawk available to rent, getting a Barbados licence is a simple process - can even hook you up with some low cost if basic accommodation if you like. It is normal practice to fly single engine over open ocean here so an adventure - this maybe the only place in the world where 20hr ppl students do their qualifying solo cross country flights 100 miles over open ocean - kind of puts the Lydd to LeTouquet into perspective - and there's no SAR helicopters here, no helicopters for that matter
