If your daughter felt more comfortable with the Tiffin 6th form then surely you have your answer? Personally, I would opt for Tiffin and here are the reasons why:
LMS has been on a downward slide for a number of years now. Just look at the year on year decline vis a vis GCSEs and to an extent A'Levels too. Look also at how the 6th form numbers have been decreasing year on year for the last three or four years also. These things surely tell you that something is gravely wrong underneath the impression the school likes to create. Have you compared the results of both schools objectively?
Add to the above some pockets of very poor teaching (I don't know what subjects you daughter wants to study) that are never dealt with and the fact that there is a high churn of staff and questions need to be asked. The last Ofsted report whilst grading the rest of the School Outstanding graded the 6th form as Good.
There have been enormous problems at this school at SLT level and this has trickled down to impact the motivation of the staff. Some very good ones are leaving this year.
A new head starts in September and I really hope she gets the school back on track and gets it heading upwards again. it really is a huge positive but if I had the choice between LMS and Tiffin I'm not certain I'd be convinced that she could make a big enough impact, especially in relation to building up a really good and motivated staff again, quickly enough for this year's 6th form intake.
The schools admissions policy changed about seven years ago and the school now is dealing with the legacy of this. It is more transparent which is brilliant but if you admit a broader cohort you have to manage it and the school has not done so.
If your daughter is an A* student and extremely bright then Tiffin would be a no brainer for me. If you feel she is not quite in that league then perhaps I would consider LMS but I think it would be a high risk decision at present. Ultimately though it's your daughter's decision and I would listen to the concerns she has already voiced - she needs to feel happy about whichever school you chose. We didn't listen properly to our own daughter because we thought we knew better. We have reaped that in spades.
Self harm, anorexia and anxiety is rife throughout most of the high performing London schools by the way and can happen anywhere. It is least likely to happen if your daughter is happy and feels in control of her destiny.