Dictionaries are right to claim it's /ɛ̃/, but the realization of this phonem varies between France (where most francophones live) and Canada (also home to quite a few francophones, but whose variety of French shows quite a few distinct features that sets it apart from the cradle of the language).
The specific set of French phonems /ɛ̃/ and /ɑ̃/ were analysed by the French linguist Henriette Walter in some interesting ways.
Firstly, she mentioned that something quite striking for a French coming in contact with Quebeckers for the first time is the way they pronounce their “an” sounds: to the French ears, Quebeckers produce something very close to the way French people produce their “in”, i.e. vent from a Quebecker sounds somewhat like vin from a French, and similarly for pairs like sans/sein, temps/teint or plan/plein.
She then goes on to explain that Quebeckers also make the distinction between “an” and “in”, but that there's a shift between the two countries. She doesn't get into the IPA, but the more or less standard way of producing these 2 sounds for francophones in Quebec and Western Canada would be [ã] and [ẽ].
Secondly, she comments that the thickest French accents would tend to shift their realization of /ɑ̃/ partway (or even a good chunk of the way) towards [ɔ̃], which is a totally different direction than what it did in Quebec (again, [ã]). So a fair contrast exists between the two countries, opening the door to the interpretation of Quebecker's /ɑ̃/ as /ɛ̃/ (or vice versa from a Canadian point of view, but francophone kids in Quebec are usually much more exposed to France French than the other way around). In reality, the ear adjusts quickly to other's way of speaking, but the first contact can yield surprises.
Back to Européen: on both sides of the Atlantic, it rhymes with vin and not with vent. One way to force the rhyme with vent, if we really wanted to, could be to compose a song for two voices performing a dialog, one of which would be given to someone from France (producing Européen), and the other to someone from Quebec (producing vent). This is a bit of a setup, though, and while it could be made credible or plausible, this would be somewhat of a novel approach to the art of French poetry.
Ultimately, there's nothing wrong with pronouncing Européen the way it is pronounced in France, even as a French-language teacher in Ontario, but the whole discourse should also show clear signs of following this linguistic norm. Switching accent based on the word would be an unusual choice (...to say the least).