No, I don't think there is enough overlap in personality between the two groups.
You can find similar threads in our pathologies - self-loathing, social isolation, traumatic childhoods, and misanthropy namely - but suicidal people differ greatly in their response to these triggers. While the mass shooter's response is to try to get even (as if such a thing could be done and as if their misguided slaying could do so), suicidal people seek an escape.
If they didn't commit mass shootings, I have to imagine these men would still cause trouble and abuse others in their lives. Indeed, many of the mass shooters have histories of domestic violence.
If somebody has a firearm they already have instant access to suicide.
Agreed. They already have all they need. Access isn't the problem here. They are motivated by much more than self-destruction.