Don't forget theres also the dI/dt rating to watch out for. Thats just as critical as I²t with capacitor dump circuits.

What's the repetition rate?

Your peak-current estimate sounds about right for a circuit with 4 uH and not much damping.
I think simulation would show an SCR to stop conducting when current reaches zero at the end of first half-cycle, leaving a large reverse voltage on the capacitor.

The SCR you pointed to looks fine for an isolated current pulse like that.   
Datasheet shows surge current limit of around 1000 amps for a half-cycle at 50 Hz or 60 Hz; your pulse would be about 1000 times shorter. 
With more details about your application, we might get into discussion about the I-squared-t limit given in datasheet.

If for example I wanted to dump a capacitor into a coil, using solid state components. Nothing too dramatic, let's say from a 1000v 3uF film capacitor. What sort of componentry should I be looking at?

Is this
https://mouser.com/ProductDetail/IXYS/CLA100E1200TZ-TUB?qs=sGAEpiMZZMvLfqGeKEshXzRZzYD%252B%2F2e1%2Fa%2F9wApXDZ3F56seD8xlLA%3D%3D
SCR suitable? rated 1200V and 160A RMS.

I'm getting about 800A peak draw(?) using 4uH load, but how tolerant would a 160A rms SCR be of this (if that calculation is accurate at all). Assuming I can run them in parallel, should I use a full 5 to make up 800A.

As I understand it electrocuting weeds works better than flaming, as the electrical heating extends right down the plant stem into the roots, instead of just burning off the top and leaving the roots intact. I've been meaning to turn an old MOT into an electro-weeding tool.

Sidenote: the circuit is grounded through a high-voltage benchtop power supply with a ground terminal that presumably goes through to the mains ground anyway but it's the secondary coil itself they're concerned about being connected to mains.

Are you sure they are concerned about mains ground connection and not just mains hot and neutral?  Most electronic equipment (scopes, signal generators, etc.) are connected to mains ground.  Circuits being tested are also connected to mains ground through the equipment ground, such as a scope probe connection.  Would you not be allowed to measure secondary current using a shunt resistor and scope?

Other suggestions include a sheet of aluminium and this is where I'd like some advice - (1a) how do I do that safely i.e. where should that sheet be kept in relation to the coil and circuit & how should it be stored? (1b) How large should this sheet be? (1c) Or are there any other safer options you would suggest (remembering this coil should be able to operate indoors)?

A counterpoise is great.  Aluminum sheet or screen or foil works well, but steel is fine too, such as chicken wire.  Easiest to place on surface such as table top, then set coil (including drive electronics) on top of counterpoise.  Electronics could be underneath, but that's inconvenient and not necessary.  Larger is always better.  Counterpoise radius on the order of height of entire coil (including base) should be plenty.  (Of course no need to be round.)
Still best to connect counterpoise to line ground besides to secondary bottom.  Likely OK without line ground connection, but I've never tried such.  Always had both a counterpoise and line ground for my coils.  Counterpoise will reduce current into line ground.

And is it stupid for the aluminium-layered box to be cardboard, given the potentially high temperatures involved?

My largest coil uses a foil-lined cardboard box.  Works just fine, even though not ideal aesthetically.  I have a scrap of polyethylene foam between box bottom and ECB, though my ECB does not run very hot.  Mylar should be good to any temperature your ECB will survive.  Or polypropylene or polyimide (Kapton) or polyamide (nylon) or many other options.

Hope your build goes well!