Isn't the queen (and Charles) anti-eu if they actually bother at all?
Ie you are betting on the wrong horse, Hrothbern ^_^
I don't think they could interfere at all, though.
I do not expect the royal houses in Europe to be really anti-eu, unless there is something personal affected by the EU. In general they, as part of the "establishment" in the EU are moderate. And that means neutral to moderately in favor of the EU.
It is also not their prime concern. I think that in good nobility tradition the preservation of their own royal family's interests in their own country will be prio 1. And usually their role for national cohesion prio 3 after their personal life with prio 2.
From what I picked up, according to royalty tea leaves readers the Queen seems not pleased with the divide in the UK, but IDK if the Queen has really an opinion about Brexit. I can imagine, thinking now on it, that her husband voted Leave and Charles Remain.
I am not really betting on the Queen to save the nation

But that element is there in the Brexit equation.
And having value for the government and part of the citizens in varying natures and degrees does ultimately means having influence. Even if you cannot catch that influence in public definitions, as is done with public institutions and public roles.
I have yet to see formal systems that work properly over a long period of time without informal systems being the practical lubricating or complementary tool to keep it going. Logical system thinkers don't survive practical managing and leading roles, however intelligent and persistent
Maybot they may be.
That influence being there, it becomes then more a matter of how you define interference, and how big the effect is.
Royalty can be compelling nasty beneath their velvet gloves. Our former Queen Juliana simply threatened with resigning to protect her hubby against something. Our PM did not want to lose votes having already a small margin in majority. The Queen while using that argument, killed immediately the arguments of all the politicians that thought that they could force the Queen with threatening her that she would have to resign.
And ofc a compromise was found: her hubby was no longer allowed to wear military uniforms, and got less pocket money from his wife
What would happen if the Queen would promise in private to May, in line with all those Ministers,
I resign, because I cannot bear having a PM that cannot unite the country ?
Will May like to live with that legacy ?
And if it would become public later on... or when May would defy it and when the Queen actually resigns, does that really give the Tabloids good fuel ?