If Timatic is considered authority, it confirms this too:
The following are exempt from holding a visa:
Nationals of Sweden with any proof of identity (other than a passport)
for a maximum stay of 14 days if they hold a confirmed onward ticket.
Multiple entry E-Visa valid for 1 year can be obtained through the
internet, prior ...
Even Rome2Rio suggests the only way is to fly to ANU (Antigua) and then you're needing to get from there to MNI.
That's an hour flying, which immediately pretty much rules out any other transport (obviously can't drive/train, so boat is the only other option and it's too far for a day trip).
Conclusion: you're going to either need to cough up for a ...
OK, so finally I actually managed to reach someone, namely the lawmakers of Montserrat themselves.
A same-day return does not count as a "through ticket", meaning that you only qualify for that passport exemption if in transit to a third country. Timatic should highlight this more clearly IMO.
So EU ID cards (except Croatian) can be used if staying for max ...
No, you would not be able to enter Montserrat using a Swedish ID card, according to a response to my PM via the Spirit of Montserrat Facebook page. Absent any other way to contact official sources, as you discovered, it would be best to carry your passport.
Spirit of Montserrat is the official Facebook page for the media entities within the Government of ...