Welcome to the club.
I'll have a go here having owned two 313PS Diesel SQ5s and now on my 3rd RSQ3.
An RSQ3 theft is just about unheard of as the model is so rare, they don't have the same notoriety as other RS models.
I'm never driven a 310, I've owned a 2015 340PS and a 2016 367PS Performance. The 340/367 models are all FL - facelift cars with the LED headlights (which are very good). When Audi upped the performance of the 310 to 340 it wasn't just a remap, the engine received changes including new fuel injectors etc - probably why Revo (and for whom I have much respect for) don't offer a remaps. There have been people on other forums that have had 310 remaps and suffered fuel starvation issues.
If I were you I'd wait for the right RSQ3, go for a facelift 340PS if you can, aside from the LED headlights there were updated dynamic indicator rear lights, a lower driver's seat, an electric tailgate and an improved parcel shelf design. Although you might aspire to the very rare (85 in the UK) performance model they don't drive any differently than the 340.
Owning.
RSQ3
The RSQ3 is an all out performance car with both a stellar engine and aural sound track to match (not synthetic) with brakes that reset expectations, those 8 pot Brembo calipers are awesome. You need the brakes because the RS's higher centre of gravity means that cornering is less competent than a hot hatch, so you blast down the straights shed the speed with the fabulous stoppers and blast out of the bend - it's addictive and really good fun. I've written many times that the RSQ3 is Audi's best kept secret. They are not cheap to own, a voracious appetite for fuel, spark plugs every 20,000 miles at £200 a set, tyres 255/35 R20, about 18,000 miles a set etc.
SQ5
The SQ5 with it's bi-turbo engine is punchy vehicle off the line, it has a box in the exhaust system that creates a synthetic lazy V8 sound. The SQ5 is large and whilst it has the pace it doesn't have either the suspension or the brakes to match the go. I have memories for sprinting (at legal speeds) between the roundabouts in Milton Keynes and discovering brake fade on the 4th island....scary. Fuel economy (actual) was high 20s low 30s - close to the RSQ3.