Hello @Ryanl
Welcome to the Wefugees community and thank you for your question!
In general, it's possible to marry no matter what your fiancé's legal status is. Your options (whether to marry in the US or in Sweden) depend on your fiancés ability to travel.
I did some research about the Swedish asylum system, but I couldn't find information about 2-year residence permits. The only thing I found is that you either get fully acknowledged as a refugee (= residence permit for 3 years) or subsidiary protection (= residence permit for 13 months):
https://***.migrationsverket.se/English/Private-individuals/Protection-and-asylum-in-Sweden/When-you-have-received-a-decision-on-your-asylum-application/If-you-are-allowed-to-stay.html
Can you maybe try to find out what exactly her status is called (is there an official term)?
As a fully acknowledged refugee she has a travel document (of course it's important to keep in mind the visa regulations that may apply). With subsidiary protection it's harder to travel because she needs her home country's passport and a visa.
In addition, the application process for a US visa can be time consuming and it costs a fee. For further information about this topic, please check out the following link:
****://***.ustraveldocs.***/se/se-niv-visaapply.asp
If you come to the conclusion that it's easier to marry in Sweden (because you as a US-citizen don't need a visa to stay there for 90 days), you or your fiancé should get in contact with the local Tax Office (Lokala skattemyndigheten) and ask them about the documents you need. They are probably the best address to go to find out about the exact procedure.
As far as I can see you need to submit a number of documents. The list includes: a valid passport, a certificate of birth, a certificate of no impediment and documents that prove that both of you are free to marry (a single status affidavit - apparently that's different from the certificate of no impediment).
Source: https://***.thespruce.***/how-to-get-married-in-sweden-2303893
I hope this answer helps you to get a better overview of the situation.
Best regards,
Thorgen