It is a bit difficult to judge really, for a start, 6 drinks? Did the mum confirm that to be true? Next, alcoholic... well shandy from the shop is alcoholic, something like 0.05%, even a slim 13yo could probably manage a full bottle or two of that without being drunk, never mind hungover.
If your DH wants to pursue this then fair enough but it is very likely she is exageratting on facebook - it is also quite likely tht you will get a hostile reception from Mum, she has obviously made her own risk assessment and decided that whatever is happening is ok, your H going in and disagreeing (rightly or wrongly) will cause waves and it sounds like dsd will be siding with her mum against her dad.
WRT drinking with you, I am a bit unclear why you have stopped this... you say she has broken trust but that would imply that your rules carry over to her mothers house, which they clearly don't (and shouldn't). She has not broken your trust just applied by a more lenient set of rules when in someone elses care - did you really expect her to say "ooh no Mum, dad and SM say only one glass of wine/shandy"?
Wouldn't it be better if you contintued toshow her a more reasonable and responsible approach to drinking? If you stop her alltogether the only frame of reference she has is what her mother shows her, if you stick to your rules and continue to explain why she will have two experiences to draw from as she gets older giving her a better chance of making a choice that is better for her.