Thank you whataboutbob, you're correct, there is no POA.
My DSIL isn't overbearing at all, thank goodness, but as I said, I just think she doesn't understand. I think she thinks that if they can diagnose an actual medical problem, then DMIL will get treatment and somehow 'get better', (even down to the dementia disappearing).
DSIL is in denial, I am just worried that the Home will do as she requests, even if they disagree. I in no way want my DMIL to suffer and she is on some medication, ABs, I just feel a barrage of tests, which will involve her being taken to hospital is too much.
But I have spoken to my DH this morning and he is going to speak to his DSIS (too far for him to go and see her, we're on the other side of the world) and explain our concerns, thank you for your help and advice.