It sounds like you've outgrown the GIO. If that's the case it does make more sense to buy a different guitar than to upgrade the GIO.
As far as the trem goes, if you're working on solos where its needed, then by all means go for it. Just go in with your eyes wide open that it's going to be something new you're going to have to learn to deal with. It also makes it more time consuming to change tunings. It's not particularly difficult, but there is a bit of a learning curve to get one set up.
I'd recommend you look for an early nineties RG570 that's in good shape. If you're patient you should be able to get one for around $400. A 550 (which is the same guitar but with a pickguard and a maple fretboard) is generally going to cost a bit more. Other than age and wear, there isn't a huge difference between the old 550/570s and the 655/657s they sell today, so I'd grab the old ones that are cheaper. If you still want to save up for a new one, you should be able to resell the 550/570 for around what you bought it for when the time comes.
For what is worth, I think everything DeeJayK has written very good advice and has an accurate perspective on the situation. Here is my advice:
You need a guitar that does the following:
1. Stays in tune (Physics will cause the strings to go out of tune over time. That is why a guitar has tuners/fine tuners. Slight adjustments are normal. )
2. Easy to play
3. Sounds good
In your case:
4. If you want single coils, you need a guitar with at least 1 single coil.
5. If you want a floating bridge, you need a guitar with a good floating bridge. (The Ibanez Edge/Lo Pro Edge are both good floating bridges.)
-This is the guitar that will make you practice more.
-This is the guitar that you will "grow into, rather than out of."
-This is the guitar that will help you get better.
Lastly, you will not play a guitar you do not like. I promise.