Looking at this from a full trip perspective, the following day trips at each city would get a fairly decent European list.
-Dublin: Full day along the coast for seabirds that would be unlikely elsewhere, including Gannet, Shag and auks.
-London: Tricky. My personal choice to maximise list length would be day trip to the Brecks for Stone-curlew, Woodlark and Bittern which may be tricky at other sites.
-Amsterdam: Morning at Vondelpark looking for parakeets, with the afternoon at a wetland site for Purple Heron, Spoonbill and Avocet.
-Paris: My preference would be for the Fountainebleau Forest for woodpeckers (Grey-headed & Black) and Bonelli'sWarbler.
-Berlin: A day trip to Mueritz National Park for Barred Warbler, Red-backed Shrike, White-tailed Eagle, Osprey and Goldeneye
-Copenhagen: Maybe a day trip to Falsterbo for migrant birds of prey and Eider.
-Prague: Probably spend some time looking for Syrian Woodpecker in city parks. Possibly Collare Flycatcher in surrounding countryside?
-Budapest: Easy choice - arrange for a day tour of the Hortobagy. Bustards, birds of prey (including Saker), wetland birds.
-Rome: Not sure this will much to species list.
-Interlaken: Probably best to take cable car to one of the higher mountains for high altitude species (Alpine Chough, Snowfinch, Alpine Accentor).
-Nice: Mornig birding the Var Estuary, and possibly looking at arranging for a day trip to Mercantour National Park?
-Montpellier: May be worth looking at doing a days guided birding around the Camargue? This post mentions a guide:
https://www.birdforum.net/showpost.php?p=3107296&postcount=2.
-Barcelona: A day trip to Belchite and the Aragonese plains could add Dupont's Lark, sandgrouse, vultures and a few other specialities.
I think 200 to 240 species would be possible with a bit of luck. For even more species, probably substituting Dubrovnik for Rome (eastern Mediterranean species), and Vienna for Prague (Lake Neusiedl and Marchegg) would increase possible species list.