It's clear that your relationship with your dp has run its course and your number one priority should be cutting your emotional losses and extricating yourself from an untenable situation. You are a capable, can-do, individual in your business life and you now need to direct your considerable acumen to your personal life.
As you know, the legal cards are stacked in your favour and I doubt that you need any advice on protecting your assets/reducing your disposable income in the event of any challenge to your status quo.
I would suggest that you rent a modest 2-bed flat adequate for your dc to stay over occasionally but not suitable for them to stay permanently for your dp and pay the rent and give him an allowance for a period of not longer than two months which will give him time to consider his work options/apply for benefits etc.
It is entirely up to you to determine how generous you wish to be in terms of the amount of the proposed temporary stipend, what sweeteners you are willing to add to the deal car running costs etc that will not affect any claim he may make for benefits, and whether you wish to dangle the carrrot of a lump sum payoff settlement at some future date to minimise any initial or residual unpleasantness.
This may come as an unwelcome shock to your dp particularly when you tell him that within two months of moving out he will need to be self-funding but, whatever or how extreme his reaction, you need to focus on the end goal which is your equilibrium and well-being.
I would also suggest that, as an additional precaution, you consider employing a full or part-time live-in or live-out housekeeper to keep your home in order and be there when your dc return from school should you be pre-occupied with work matters.
With regard to your dc, I suspect that they cannot be unaware that all is not well between their parents but, handled properly and with consideration given to their wishes with regard to seeing their df on a regular basis, I see no reason why their lives should be unduly disrupted or that they should suffer any unnecessary trauma.
With regard to your wider family, friends, the local community etc, your changed circumstances may be a five-minute wonder but dust settles quickly.
The loss of your mother has undoubtedly taken an adverse toll on your physical health and grief can manifest as overwhelming exhaustion and despair (as I've not read any other thread you've posted I am unaware of what impact your dp's behaviour during that time may have had on you). As you've observed, the old adage has it that we should make no big alteration to our lifestyle for at a least a year after the death of a loved one.
However, it occurs to me that as much as you are grieving for your dm and the security that comes from having an older loved one in your life that 'knew you when' and all of the things she meant to you, you are also grieving for the loss of a meaningful relationship with your dp - albeit that any meaningfulness you once shared may have revolved around your dc, or that it may have been a somewhat pedestrian muddling along rather than a great love or meeting of minds.
On this basis I see no reason why you should not divest yourself of the millstone around your neck asap as you will float to the surface and inhale the welcome relief of less complex air.
The same may not be said of your dp who may be in need of his scuba diving gear if he cannot adapt to the dry land of reality. Should your dp appear to be in danger of drowning you are best advised to resist the temptation to dive in and rescue him.