Alasdair, it's not just a factor of budget, there are just few that sit in that intersection regardless. Probably those that would do both well would be a large apo refractor, but they're serious money by any metric! Not to mention the size and weight, and the mount required to carry them.
I guess what I'm trying to get across is not to be discouraged by budget. You can see amazing wonders and take surprisingly decent images - and most importantly, have a lot of fun - with much more humble equipment. Most of us fit into this category
To date I've done most of my beginners imaging with a 71mm doublet refractor that I originally bought for chucking in the backpack with the laptop when travelling, and I used it in a camera tripod for visual' Does it have limitations? Sure. Have those mattered to me? Not so much
For extra kicks, I do still occasionally use it for visual, and it gives a mega wide field of view, enough to swallow Magellanic clouds!
So yeah, pick the direction you feel most passionate about to get started and you can always add more later. For visual, a 8" f/6 Dob is hard to beat for combination of price, performance, what you'll see and practicality. A 10" scope is substantially bigger. If you get some free time in the city drop in to Bintel and have a look, the difference is all the more apparent in the metal. With your budget you'd be able to pick up a scope and some essential accessories, like collimating tool and a maybe another eyepiece or two.
Also, when you've looked at objects from a city location, take your scope out for an evening out in the country (where safe to do so) and you'll feel like you've got a bigger scope for the cost of the fuel