A few misconceptions here. Firstly, if the story is true (I've no reason to doubt, I just haven't actually seen it though!) then Kanye will not be suing Lloyds - as it isn't an insurance company - it's a marketplace for syndicates of underwriters to transact insurance business. Secondly, the syndicate in question won't have to prove Kanye's mental health or otherwise - the policy will most likely be written on terms that respond to the happening or otherwise of a particular incident (the insured perils). If the cancellation was not as a result of one of these insured perils, the policy won't respond. Perils would be storm, civil unrest, equipment failure, physical injury and other similar ones. If it's not specified, it's not covered.
The burden of proving it falls within cover is on Kanye - so, say he claims illness as a reason for cancellation; firstly they would check whether it's covered, and if not, they've reject his claim. It all depends on the terms he agreed beforehand. So if cannabis psychosis (or other mental health issues) are not within the scope of cover, the policy would not respond.
Noone's using mental health as an excuse or stigmatising it.