The definition you cite restricts this meaning to “legal, formal” language. You will find it most often in adversarial contexts such as legal or political debate, overwhelmingly in the present tense as part of an argument:
I submit that this case is in the nature of an action of trover, to recover the possession of this wig; and this admitted, Sir, I have humbly to contend, that the plaintiff must be nonsuited; for, Sir, you will not find one word of or concerning
a wig in his declaration. — The Sydney Gazette and New South Wales Advertiser, 19 Feb. 1824.
We submit that any such animal will remain human imprinted, even it is "rehabilitated" as proposed in the planned legislation. — Chris Mercer, Beverley Pervan, Canned Lion Hunting: a National Disgrace, 2005, 31.
Claimants submit that the standard for expropriation is objective and that while the showing of intent to expropriate may evidence a measure to be expropriatory, it is not a requirement of expropriation. — Kaj Hobér, Joel Dahlquist Cullborg, Investment Treaty Arbitration: Problems and Exercises, 2018.
When someone submits that something is or is not the case, they are not merely saying or suggesting something, but asserting whatever comes in the that-clause as a true fact or accurate description of events, almost always in opposition to some other view. And they are doing so only in particular rhetorical situations: in a court of law, a legislative body, a formal board meeting, and, as in the book on lion hunting, before the “court of public opinion.”
It would thus be highly unusual for submit that to be used in the context you suggest.
