LI are Danmachi only while she can have a relationship in PJ verse it can't be with a Goddess though it can be one sided for the mentioned goddess.
Mc x Freya x Hephaestus x other goddess or long lived person
Mc can be reincarnated from our world or not into Percy Jackson but she must have basic knowledge of danmachi even if it only happens when she is transported to danmachi verse as a perk or it's manga/anime is out during the Percy Jackson verse timeline.
She can be transported to Danmachi verse anyway but she must be thought of as died in PJ verse though you can have the PJ verse find out she is alive but she can't come back though they can watch her just not all the time so for them it's a bad reception.
It doesn't matter who she is the Daughter of but by the time Danmachi comes in she must have blessings of the other gods this is mainly for the Danmachi gods reactions so it will be a double whammy for them cause our girl is the impossible for them since it's cannon for the danmachi gods can't have kids.
This lets you play around with Mc having god blood plus blessings that are part of her essence/one with her so Mc is either immortal or extremely long lived because of different world rules each level up she gets closer to god hood by level 5 she is a god but she can keep leveling to level 10 she just gets stronger.
It doesn't matter which familia she joins doesn't even have to be a cannon one because of the above it has to be someone she can trust cause she will start with a skill that states the above for her ichor/god blood.