And not only that but he would still be in the line of succession... (with a small but still real chance of being called to the throne if something would happen with William and his family - in the past we've seen whole (royal) family units die in an airplane crash, which I sincerely hope won't be repeated but an accident/crash of this magnitude is possible for each and everyone of us). However, the British apparently don't really care about foreigners in their line of succession as their line of succession even includes foreign monarchs...
And there are issues with Harry's (and possibly Archie's) role as Counsellor of State. The criteria include
Councillors of State must be at least 21 years old (except the heir apparent or presumptive, who need only be 18 years old), they must be domiciled in the United Kingdom, and they must be a British subject
The word "domiciled" is the key. I'm not quite sure how the Privy Council, particularly government secretary of states would react if a Counsellor of State who don't permanently lived in a UK, can preside over them and "represent" the sovereign. Unless of course, he/she was very informed (by extensive research or advised by royal staff).
From Wikipedia, I know it's not accurate, but gave examples where the Counsellors of States are not in the direct line of succession, not technically senior working royals and not hold Royal styles and titles.
Since the passage of the Regency Act 1937, the only persons to have been Counsellors of State while not a queen consort, prince, or princess were George Lascelles, 7th Earl of Harewood; Alastair Windsor, 2nd Duke of Connaught and Strathearn (although Windsor had been a prince between 1914 and 1917 and never served in practice during his short tenure); and Maud Carnegie, Countess of Southesk (who was entitled to, but did not use the style of princess).
Unlike Harry and Archie's case, 7th Earl of Harewood, Alastair Windsor, 2nd Duke of Connaught and Strathearn and Maud Carnegie, Countess of Southesk (legally Princess Maud, Countess of Southesk) mainly resided in the UK.
The question isn't about having someone in the Privy Council who is not a working royal, but more about permanently reside in the UK.
It would not surprised me if some Privy Counsellors prefer Princess Beatrice as a Counsellor of State over Harry and possibly Archie, despite being lower in line of succession and was never a working royal. This is assumed that Princess Beatrice remains living the UK.