Employer's Broad Ip-assignment Clause Vs. Software I Built

Location: Canada
I've been a full-time employee (~4 years) at a company in the natural resources sector. My employment agreement has a broad IP clause: any invention, improvement, or process "relating to or usable in connection with the business of the Company," made during employment, automatically becomes company property — plus a standard confidentiality clause covering company know-how, methods, and technical information.

Over roughly the past two years I built an internal software tool related to a technical process I handle as part of my job. An early version was started on my own time, on my own computer, before it became a more active project; since then it's been developed further using company time/resources and is now used operationally.

Recently, a third-party contractor that works with my employer approached me directly wanting to buy a one-time license to use the software for their own purposes. My employer doesn't sell software — it's not part of their business — so I don't believe they have any existing plan to commercialize this themselves.

I haven't agreed to anything with the third party, and I plan to disclose the tool's existence internally before doing anything further, per my contract's invention-disclosure requirement.
Questions:

Does the "own time, own computer" origin of an early version carve out any personal ownership, or does the broad "relating to the business" language capture the whole thing once it was continued during employment?

Is it realistic to ask my employer to formally license or assign rights back to me, given they have no software business of their own? What's the right way to propose that?

Does disclosing that a third party already wants to buy it help or hurt my position when I raise this internally?

Not asking anyone to draft anything — just trying to figure out if this is worth a paid consult with an employment lawyer before I say anything to my employer.