I suspect there are going to be plenty of bargains around in May! However if there is a coronavirus epidemic ‘abroad’ (wherever that fictional country is!) there will probably be an epidemic here too. I do think £300 for 3 of you is ambitious if you are planning on flying anywhere though. You can find flights with Ryanair for about £50 return per person to a number of cities I would consider taking the kids to, and that seems cheap but has already used up half your budget. It is also possible to find cheap hotels for about £50 per night for a room with a double bed and sofa bed. They are not the most luxurious but make perfectly good bases for adventuring. This leaves you with about £50 for all food, other travel and entertainment. Some cities you can have a fab time walking round and doing only free stuff - Rome is the most obvious of these, with the pantheon, Trevi fountain, Spanish steps all being free and lots of other things you can marvel at the outside of without worrying about being a skinflint - the colosseum, piazza Venezia, Vatican City, all look amazing even just from the outside. There are other cities I’ve walked around that are just as amazing without paying to go in anywhere too.
If I were looking to spend only £300 though I would be more inclined to agree with the posters who suggest camping and book a ready pitched tent or glamping pod. Take your own food, some maps and have an outdoor adventure with a couple of pub lunches and lots of making hot chocolate and marshmallows over an open fire whilst playing cards.
May is not too cold to be doing this if you take enough clothing. I camp with my kids every year and it’s one of the highlights of the year to them. We go to Badminton Horse Trials where a ticket for the entire week for a camping pitch and 2 adults (kids go free) is only £200. Admittedly you have to bring your own tent for that, but the atmosphere is fab, especially with this year being an Olympic Year and so many riders will be bringing their best horses to try and secure a place on their countries’ team. There’s a fairground, restaurants, bars, a big screen to watch the arena events on if you don’t get arena seats (they are pricey for the showjumping but free for the other days,) and the biggest outdoor shopping village in Britain (so scope to spend too much money there!)