A bench PSU's transformer *NEEDS* either
reinforced insulation between the primary and secondaries, or basic insulation either side
* of a grounded interwinding screen. Simple basic insulation isn't sufficient as for a bench PSU, the secondary wont be grounded. Your transformer is almost certainly not built that way as the presence of the wire internally connecting the primary and secondary sides indicates its likely to have no more than functional insulation.
The *ONLY* way to use it safely in a bench PSU circuit would be to fully rewind at least its secondaries to implement either a reinforced insulation barrier between the primaries and secondaries, or two basic insulation barriers with a full width grounded foil interwinding screen
#, with proper creepage and clearance distances (which would require either a bonded built-up bobbin to achieve the creepage distance requirements at the edges of the winding, or rewinding the primary to implement margining), with significant loss of winding area and thus VA capability. You'd then still have to soak test it at full load and Hi-Pot test it at 1KV to see if its actually fit for purpose.
If I were you, I certainly wouldn't want to tackle the job without a well equipped transformer winding shop and prior experience or a mentor, to guide me away from the possible safety pitfalls. Also its *NOT* worth doing unless you need to reproduce a transformer with an unusual custom set of secondaries, and cant simply order one from a custom transformers specialist.
TLDR: Its not fit for your purpose nor can it be made so without several times more materials and labor than its worth.
* In theory you only need functional insulation between the screen and the secondaries, but that would be unsafe if the supply ground was ever lost.
# The foil screen must *NOT* form a shorted turn. It needs an insulated overlap of at least the minimum creepage distance. Also bringing out the ground maintaining basic insulation between it and the secondaries may be problematic.