How far are you willing to commute? Your relocation agency will probably not even mention to you that there are, shock horror, loads of places to live off the island! They will also not show you anything low cost (not that you may want this). However, if you want peace and quiet, you will need to travel. But it CAN be achieved in Hong Kong. Could you convert your housing allowance into a mix of housing plus a car allowance? Because if yes, then plenty of places within 30 mins drive of Central are then opened up. If you don't have a car, those places are perhaps 10-30 mins further away (depending on exact location and choice of transport). I live in Gold Coast, which is quiet and where 28k would get you a huge place, but it's 30 mins (by car) or more like 45 mins by public transport (shuttle bus plus MTR). I have no clue if we have a gay scene here. Closer are places like Sham Tseng which has a direct bus to Central taking around 30 minutes. That also has a nice ambiance, beaches, restaurants and an international supermarket. You might also consider Park Island, which has a direct Ferry to Central taking about 25 minutes. It's a car-free place and many who are looking for a more relaxed weekend swear by it (other says it's lacking facilities - one man's peace is another man's hell!).
You could also look in the other direction - on the South of HK island. Places like Stanley for example may suit (note that Stanley fills up with tourists every weekend though). Or even Sai Kung area - another lovely place to live well within your budget, and with it's own vibe and many expats - but not particularly conveient for Central depending on the exact spot and route. That also gets touristy at weekends, but in a manageable way mainly.
Best bet is to get a serviced appartment for a couple of weeks and travel around here and look at the districts. Don't make any decisions before you arrive. If your relocation consultant is the "typical" type, I would ditch them soon as. But if they appear to be actually helpful and capable of thinking outside the box, at least they might be able to drive you around! (Having said that, using public transport gives you a better feel for how far out places are in practice).
Good luck.
This post is not to say ignore HK island, but just to highlight that if you ARE really looking for quieter weekends there are options, and actually, you pay less for peace and quiet, but you do pay for it in commute. Your choice.