Don't know about the impact on your particular school admissions situation, but I can say with confidence that there will be no long term problems arising from skipping SATs. Many home educated children go to school at some point in secondary education, with no SATs, and I'm not aware of it ever causing a problem. 4 of mine have, and through years of attending home education groups I've known plenty of others in this situation. Not having SATs has simply not been a problem for the children.
The primary purpose of SATs is a school performance measure. They are used for the baseline measure of Progress 8 etc, but again, that's a school performance measure rather than a pupil performance measure. It's of no consequence to the child - it doesn't go on their CV.
GCSE predictions are made on the basis of mocks taken during the GCSE course. There are provisions in the school accountability guidelines for the situation where SATs are not available ; that child is excluded from the school's Progress 8 figures, but is included in the Attainment 8.
The only case I can think of where the absence of SATs might have had an impact was a formerly home educated child who started secondary at a school where there was strong streaming from the very start, based on SATs. The mum was proactive about it and spoke to the school, said she didn't want her child just placed in a middle set because they didn't have any SAT scores to use. Clearly they found another way to test the child as he was put in the top set and has since done well.
The end of primary can be a nice time to try home education - knowing it's only likely to be for a few months, you can really go for it and do lots of fun trips. It would be a shame to miss out on the end of primary "graduation" stuff though. You could be cheeky and deregister her for a few months, have (hopefully!) a lovely experience, then re-enrol her at the start of July I suppose. The school census has just passed so the school has already presumably got a full year's funding for your daughter anyway.
Can you email the secondary you've applied to and ask what they would do of an applicant was not taking SATs? They must have some provision to deal with the situation.
Good luck!