classic motorcycle forum

Always puzzled when people talk of parts being hard to get. About the only awkward thing on that Cov Eagle might be the cast alloy pong box or possibly the piston but what else? It’s not exactly as if the bike is high tech aerospace engineering. Plenty of New Villiers stuff around.
There’s plenty of information available. Cov Eagle were good with illustrated spare parts lists and both Villiers and Albion Info is plentiful. What’s the problem?
Sorry to sound grumpy but a little research will pay big dividends.

Every country has its share of bumf loving bureaucrats. The trick lies in how you deal with them. Unfortunately sometimes you can’t avoid them.

The classic example is the English DVLA. Just look at all the rubbish that they generate. The rest of the world isn’t interested but every bike forum gets stuck with their rubbish sooner or later. English insurance runs a close second.

Usually found more in the veteran car world. Last one I saw was in a veteran Silver Ghost Rolls-Royce tool kit. I suspect it may be a standard touring kit item.
Easiest place to start would be Hunt House, HQ of the Rolls-Royce Enthusiast Club. You may even find a certain degree of interest from them if your replica is up to snuff.

If I remember rightly the 3T frame was smaller than the standard frame which led to it being used on the early Trophy. I can remember a mate going on ad nauseam about this as he built up a 3T which he assured me he would eventually build up into the definitive Trophy.
He only built the 3T because he got the engine at a knockdown price. 3T’s were viewed with contempt back then. Trophy engines were a lot dearer so he never did do the changeover. Toms wallet never came out in daylight anyway so none of us were surprised.
So, Brad I’d check what frame you have before leaping in.
Finally, I don’t know what part of Oz you live in but all this happened down in Victoria. No problem getting Triumph stuff then.

Probably are.
Takes a while for new owners to appreciate you have to ride the bike for quite a few miles for both the engine and the complete exhaust system to get up to operating temperature and blow out the excess oil. They load up surprisingly quickly in stop /start traffic or if only ridden commuter distances. Will then need a good ride to blow away the cobwebs and may lay down a Royal Navy quality smoke screen for a few miles while doing so.
The effect is magic. De-clagging the exhaust gives much the same result as a major engine tune up. You really notice the difference (and the engine stops smoking).

It is a nice and clean looking jigger isn’t it. If that little tinware is missing and you have the other side as a mirror image pattern it shouldn’t be so hard to have made what you need.
You could approach the AOMC for a look at the old Sun rego cards. There can’t be that many of them to shuffle through. May have some useful info. It will certainly make clear what came into Victoria.
The UK VMCC library has a selection of possibly useful Sun material but they say a lot of it consists of poor photocopies. Better than nothing I suppose.
Finally there is Morton’s. Your lower Overlander photo appears to have come from an old Motorcycle or Motorcycling roadtest. Morton’s will have the original plus probably all the photos taken for the roadtest. Could be useful for different shots of the tinware. Worth a try. They may have other Sun material as well.
Very last thing. You could spend an entertaining afternoon in the State Library going through their copies of MotorCycling. It takes willpower to churn through eighteen months worth of magazines in an afternoon without being distracted by interesting articles but doable if you’re determined.

You're complicating a very simple issue.
Your Sun can be identified by both the engine number and frame number. The Century was a one year only model in 1957 when Sun was thrashing about in its final death throes. Give us the details and we'll tell you more.
As for a parts book, you don't need one. You bike is built to the standard British formula. You have a Villiers power unit including lighting, British Hub Co wheels as used in the majority of British lightweights of the day, an MP or Armstrong front end and proprietary rear suspension units (probably Armstrong). Very little was actually made by Sun. Only the frame, tank and seat. (The seat base on my mates Sun Cyclone had been stamped out of an old WWII ammunition box. We found out when he got it refurbished.)
To the best of my knowledge most Sun of that era were imported in to South Oz. Certainly none to Victoria after 1954.
All in all you have a typical British lightweight of the period which will serve you quite well and stands out from the usual pack of James and Fanny Bee's.

I’d suggest a clear photo would help.