Given the nature of the times would they really choose another woman though? Three female rulers in a row
Arbella is a woman in her 20s who would, unlike her predecessor, not throw up when someone mentions marriage within a 10 foot radius of her. There's still going to be a king, and a very big chance of an heir soon after. Marrying one of the male claimants further back in line could unify multiple claims and increase support. Or they can just marry someone without a claim to keep the other claimants' parties calm.
The business of the marriage with Seymour in the last months of the reign of Elizabeth I and the reactions which they arouse can precisely push the queen to choose the young Henry, by planning a regency, than Arabella who can appear to her more intriguing and unstable?
I don't recall anything actually happening when those rumours arose?
Also, to go back to the case for Henry Frederick, by the time Elisabeth dies in 1603 Henry would already have spent +/-3 years under a
Scottish regency. And this "foreign" (to the English) influence over him definitely isn't going to help his case.