In an electronics workshop - which might be slightly different to what the OP is doing - then there is only one use of an isolation transformer: to allow the safe connection of test equipment to a DUT. The classic example has already been given; the primary side of a switched-mode PSU (where the -ve output of the input bridge is a negative-going half-wave waveform wrt mains earth with a peak value of -325V). In days gone by, prior to baseband video/audio inputs, TV sets had a live chassis, so also needed to be isolated for the same reason.
Note that "allowing safe connection of test gear to the DUT" is nothing to do with the probability or magnitude of electric shocks - the transformer is not intended to change that. Naturally, as soon as you've connected test gear, the DUT is no longer floating, so it remains hazardous; but the transformer has allowed you to tie it to ground at the point you need to make a measurement...
A really important rule - not yet mentioned, I think - is this: only one DUT may be connected to the transformer.
Also, the mains earth should not be carried through to the outlet. If you have a DUT with a N-E swap, the case becomes live.
The default working practice it to power a DUT from an RCD, and only use the isolation transformer for those occasions described above. Naturally, when using the isolation transformer, any RCD in line becomes redundant, but doesn't hurt.
Away from all that, if you wish to use an isolation transformer to reduce the magnitude of an electric shock, then you need a centre-tapped secondary, and you ground the centre-tap. This gives you two lines at 115V wrt mains earth. And you should be able to use an RCD at the output (but check it works properly in that mode). But naturally, this supply is no longer floating, so can't be used to safely connect test gear to a DUT as described above.
Personally, I doubt the utility of this arrangement in an electronics workshop environment, and have never used it myself, but it might be appropriate for the electrical repairs that the OP does? Of course this scheme is used on building sites - those yellow-cased "site transformers" produce 110V from a transformer with a grounded centre-tap, so the maximum voltage wrt mains earth is 50V.
I've attached a BBC EGN which might be of interest.
All the best,
Mark