yeah, just file (I don't actually understand how you do mediation without filing anyway, my ex filed and then we did mediation).
Put everything in writing. Ask for a little more than what you're willing to settle for (negotiations are business deals and you always want to leave some room for the other person to feel like they won something. In your case this might mean a short delay on selling the house instead of putting it up immediately, or 40/60 custody instead of 50:50, or something to do with debts or pensions).
Make it clear her "safety" concerns are unreasonable and you'd be happy to have BOTH of you evaluated by a psychiatrist (I'm guessing she'd be easy to detect as a looney tunes, and any psychiatrist would laugh her out of the office if her main complaint is you grabbed your kids leg once and yelled at him).
If she doesn't have an order for temporary support, you need to start paying her only what she would get in a divorce. Judges will listen to the idea of "imputed income" - what she could reasonably earn if she got a job. Right now she has no incentive to change anything. So - pay the mortgage - yourself, don't give that money to her - and then pay her what your state's alimony calculation would likely be between her imputed income and your actual income. You could subtract half the mortgage payment from that. Then add in any child support you might owe if you had 50:50 custody. I hope you've already taken half of any savings but if not, do that now.
You need an attorney separate from the mediation lawyer, if you don't already have one, get one now. Even if you continue mediation, they can be your sounding board and tell you if what you are agreeing to is reasonable and what a judge would give you in court. Get him or her to review your proposal before you make it. Remember too that the tax laws in the US have changed, alimony is no longer deductible to the person paying it. Work out an offer with your private attorney and then present it in mediation. If her response is totally unreasonable (which it sounds like it will be) then you may be better off filing and let the lawyers duke it out. Mediation only works when both people are at least somewhat grounded in reality.