I also don’t have a car, but as I’m a) a pensioner and b) live in Wales, I can get about free on buses. Looks like a lot of small parks and public spaces, and the airport FRZ; I see what you mean. This means you need to think outside tbe box when you look at maps!
Assuming you can use public transport and are ok on your feet (sorry, I know it’s a stereotype, but, you know, “Bournemouth”), there seems from a brief squiz to be a large area of open ground in Christchurch, in the Avon Valley north of the railway station. Power lines on the west side and trees on the east, but a big space between.
If you are bothered by dogs or birds (seagulls and, especially, oystercatchers will try to ‘mob’ your drone), switch to ‘Sports’ mode and fly directly upwards, vertically at full throttle; a dog can’t jump more than 20’ at most, and I’m talking Doberman for that, and birds cannot fly directly upwards at anything like your speed. Once you’re high enough, get outta Dodge, fast!
The beach gets less crowded towards Christchurch as well, and, if buses can be used, there is a lot of open common area on the Purbeck Hills and Kimmeridge Ledges. Hengistbury Head looks a good spot, too, and there’s plenty of open heathland around Studland! NT does not allow TOAL from their land, but you can overfly it from a public road…
SSSI’s are not ‘out of bounds’, they are marked in yellow on Drone Scene maps as ‘Ground Hazards’. That means that they are features that you need to be aware of as a pilot, that might need particular care to be taken when you are flying there, and as well as SSSI’s they include tall buildings, masts, cables, railways, motorways, and ‘sensitive’ locations like hospitals and schools.
It is of course nesting season, and if the SSSI is designated for bird or animal reasons, then you need to stay away from nests and burrows, but unless you deliberately buzz the wildlife (in which case you deserve anything that’s coming to you), you will do no harm. All SSSIs are highlighted in yellow by default, even when they are because of geological interest or because of rare plants; flying over rare plants won’t affect them (but watch where you are treading)…
Recreation grounds and public sports areas are good as well. Probably not on a Saturday morning when all the local league games are being played, with spectators lining the pitches, but midweek they are likely to be quieter. I use a recreation park about 10 minutes walk away for practice and ad hoc unplanned spur-of-the-moment sessions, and at most times there are people about, dogwalkers included, but not many and not a problem. Keep high enough for the noise not to bother them and most of them won’t notice you’re there, and you’ll be safely above the trees*! In a year of flying I am yet to have an aggressive reaction from anyone; perhaps I’m lucky! In fact the main problem is people being too friendly and wanting to know all about the drone, which is distracting while you are flying. I just put the drone into a hover and chat to them!
*Give trees at least twice as much room as you think they need. They are tricksy little buggers, and will bring your drone down at tthe least touch, as well as being able to move in the wind. They’ll jump out on you when you’re not looking.