We have managed on a similar budget in school hols last few years. We dont book last minute though. We book Easyjet, hand luggage only (no weight limit but dimensional limit) as soon as flights come out. Typically we pay under £30 each way with seat reservations. Accomodation is typically £50 - £100/ nt for something 3/4 bed with reasonable facilities. The change in the Euro exchange rate and inflation has ment that the amazing house near Copenhagen we stayed in with swimming lake at the bottom of the garden was about £700 is now over £1000. Airports are now giving less than 1euro to the pound.
We self cater generally air bnb/ homeaway/ bungaloo. We like somewhere spacious with wifi.
Without car hire we've been to Venice, Prague, Berlin, Vienna, Geneva and Chamonix (flew Geneva and easybus to Chamonix flight plus bus around £45 each way) with hire cars Iceland and Denmark. We went to Iceland again this year and found it prohibitively expensive this time so it'd no longer be on my recomendation list. Denmark was wonderful and different. Car hire for a really nice from airport car was around £20/ day so a great price.
Lots of Northern Europe August is mid or even low rather than peek season so getting cheaper.
If you can do flights for around £300, accomodation for £500 (£50/nt) £200 transfers then its food and activities left to budget for. I tend to budget £80 - £100/ day. A simple activity like swimming for 5 can easily be £40, a quick drink after and then food and the days budget is gone. Theme parks could easily be £200 for a day. So looking for accomodation that has activities that are affordable, or on a site with stuff included can be worth factoring in when choosing your accomodation budget.
Its doable if you really budget but spreading the budget over 7 nts rather than 10 might give you just a bit more money to enjoy your holiday
Have you factored in passport costs, insurance etc?