After flushing a newly installed toilet, the water will not remain in the toilet bowl but no leaks are present

Water is not draining completely from the toilet line to the sewage tank. If there is a 'low/little slope' section anywhere in that drain line, solids WILL settle and collect there and cause water to back-up in the toilet drain line. Then the toilet will not flush completely or at all until the 'soft blockage' is cleared out and the drain line emptied. Al 'gravity drain' toilets require a free draining/empty drain line that allows flush water to drain unrestricted. A basement toilet has a pump that 'forces' flush water up and out with enough pressure to sometimes force out the 'soft clog.'

Look inside the tank under the lid, it sounds like the flapper that lets the water into the bowl is getting hung up on something when you flush. It could be the flushing lever. This would cause the flapper to stay open letting water to just pour into the open sewer line. It should be a simple fix. Just see what is binding the flapper & make an adjustment.

Fixing a leaky toilet is actually rather simple, requiring few tools and just a little time and effort. In looking at Caroma toilets on Google, they appear to be just a standard type of unit, and should present few if any problems for yourself or any handyman of modest skill.

If it is leaking around the base of the toilet - where the toilet sits on the floor, the repair item is a simple wax seal. Here is a tutorial to effect this repair: Replace Toilet Wax Ring. The wax ring is very inexpensive, usually $5 to $10 USD.

If the water is leaking from between the tank (cistern) and the toilet bowl, the job is even easier, as it does not require lifting and moving the entire toilet. A good tutorial for that repair is here: How to Replace Tank to Bowl Gasket . The gaskets, new nuts, bolts and rubber seals come in a kit and also cost only $5 to $10 USD.

I am sorry to hear your Plumber insisted you buy an entire new toilet, but do remember he is in the business of making money!

The bowl is cracked and is leaking down the drain. You will have to replace it. It may have cracked if the bowl hold-down bolts were overtightened, or the bowl was not able to sit flush (pardon the pun) on the floor. If the bowl rocks when it is first installed, instead of tightening the hold-down bolts, use a couple of tapered shims under the edge of it, then caulk to fill the gap and hide the shims.
Good luck.

turn off the water,, flush the toilet ,, using a rag or paper towels sop up the remaining water in the tank, ( if you hold the flush valve down more will exit the tank) remove the water inlet tube, remove the 2 or 3 hold down bolts that have a slotted screw head inside the tank, and a nut on the underside of the bowl, use care in lifting the tank off the bowl section, as they are normally a matched pair and most aren't interchangeable, on the underside of the tank you will see the bottom of where the flapper attaches,, at that point will be the gasket that needs replacing, remove it, and install the new one, reverse this order to re assemble,, NOTE, you don't have to tighten the tank bolt very much,, snug is better than over tightening, the tank will crack if you over tighten the tank back on the bowl, so use care in this step, I always replace the tank bolts at the same time,

Sorry for problem with new toilet.
You say water is leaking out of the bowl?

Toilets have 2 parts:
1) Tank: where water is stored for the flush and is sealed with a flapper.
2) Bowl: where water flushes down.

a) If tank is leaking water, then flapper is suspect. Replace flapper and see if that works.
Lift up flapper and run finger around rim to see if there is a rough spot that would indicate flush valve is defective, and needs replacement.
See typical flush valve

b) If bowl is leaking, that is very unusual and would indicate a crack in the porcelain.

If installer sold you the toilet, it is his responsibility to make good on product.
All products are guarantee against workmanship flaws.

replace the flapper

maybe by shortening the chain length on the flapper valve might help

2 possibilities.

1) The diaphragm in the Sloan is for a urinal. (1.0gal vs. 1.6gal);

2) Not enough water. (stop partially closed, supply undersized)