FLR(FP) is an exceptional circumstances visa which is granted outside the immigration rules in the case where the applicant does not qualify for FLR(M), but they have a strong case under Article 8: Right to Family Life in the UK. If UKVI determine that they should be allowed to stay in the UK under Article 8, they will be put on the 10-year path to ILR outside the rules, instead of having their visa refused outright.
If they do not have a strong enough case under Article 8, their visa will be refused, they will have 14 days to leave the UK, and all future visa applications will be automatically non-straightforward
In order to have a strong case for Article 8 and an approval under FLR(FP), you would have to prove:
- your partner has a UK citizen child living in the UK whose life would be severely disrupted if they were separated from their mother
- your partner's life would be in danger if she was forced to return to the US (i.e. due to war, famine, terrorism, natural disasters etc.),
- your partner has no living family members or means of support in the US, which would mean she would end up homeless and alone if she had to leave the UK
- you are ineligible to enter or are banned from entering the US (i.e. due to medical conditions or criminal convictions) which means you will never be able qualify for a visa to live there with your partner
- you and your partner will never be able to qualify for visas to live together in any other country in the world, and therefore the only way you can be together is if she is allowed to remain in the UK
FLR(FP) is not something you should be hoping for or aiming to get. It's a discretionary visa that is only granted to someone who cannot qualify for the proper visa. It's also more expensive because it doubles the time you have to spend on visas before you can qualify for ILR or citizenship (10 years instead of 5 years), and it means paying for 10 years of NHS use instead of 5 years, plus her not being able to qualify for UK benefits for 10 years instead of 5 years.
You'd likely be much better off with her either getting sponsorship to switch to a Tier 2 visa from inside the UK, or her returning to the US and living there until she can qualify for a visa to come back. It would save money on visa fees and could save several years of time spent in the UK on visas.