Capitals :

[Josef Essberger]

Is batman a rank?

[Josef Essberger]

My own feeling is that this is a matter of style and convention rather than grammar or "correct usage".

Generally there is no need, for example, to capitalise prime minister unless it refers to a specific prime minister:

He observed that Prime Minister Truss was the Queen's sixteenth prime minister.
How many queens has the UK had?
Is it true that President Biden is the oldest US president?

And this (opinion) just in from an ex-serviceman friend and writer:

The sergeants of the Household Division, or captains in the British Army in the Napoleonic era, where they are being referred to generally...
And to answer your actual question, it just does not sound right to be Batman. He would be 'Private White, batman to Captain Black'.

[Josef Essberger]

To style and convention I would add context. So things might depend, for example, on your readership (if they are used to a particular style of a given regiment).