Is there a safety issue with adding a grounded transformer to an ungrounded network?
The lack of clarity in your question would be greatly reduced with some diagrams.  In most situations, connecting transformers to ground simply sets what 0V is in relation to each other.  Each voltage source can produce positive or negative voltages at any moment in time in relation to ground or 0V.
In a well designed system, the voltage of ground will only differ significantly from point to point when massive ground currents are flowing, which should trigger a shutdown, whether blown breaker/GFCI or actual generator shutdown.  This allows us to just assume ground is 0V.
Transformers perform electrical isolation, so having the secondary grounded differently from the primary is not a problem.  If electrical isolation breaks down, you could see full primary voltage on the secondary, so then you have bigger problems than where your ground is attached.  In this case the transformer has been destroyed, typically resulting in a large fault current to trip the upstream overcurrent or ground fault protection.  Depending on how the fault occurs, primary voltage could appear on the secondary for long enough to damage secondary systems.
When split single phase is added to a 3 phase system, sometimes the primary is connected to only 1 phase of that system to achieve &quot;true&quot; split phase where the positive and negative voltages directly oppose each other in time rather than the \$1:\sqrt{3}\$ ratio achieved with a 3 phase wye arrangement.  Advantages include having fewer wires to the primary and simpler wiring on the secondary and a more useful line to line voltage.
As soon as you start putting single phase loads on a 3 phase system, load balancing becomes an issue.  Ideally load imbalance should be small compared to the total power of the system.  As long as there are a sufficient number of panel sized single phase loads that they can be spread across the phases of the system, there is no advantage to running 3 phase to all of the smaller loads fed by the panels.

What actually happens when one of the phases that feeds this
&quot;single-phase&quot; transformer goes to ground?

A large current flows from the source feeding the primary, tripping the upstream overcurrent or ground fault protection device.  The ground reference of the secondary is irrelevant in this case as the fault will shunt current away from the primary winding feeding the secondary, and the current is flowing back to source.

What voltages will present themselves and in potential to what?

All non-isolated voltages will present themselves as positive or negative at any moment in time in relation to ground.  One of the coils on the secondary will be in opposition to the primary and one will be in agreement with it.  If you cross connect the systems line to line, the highest voltage you could get would be a line to line sum of the line to ground voltage of the primary (460V / \$\sqrt{3}\$ = 265V) and the line to ground voltage of the secondary (presumably 240V / 2 = 120V).   The resulting line to line voltage of 385 is less dangerous than the 460V line to line of the source system.  Note that in this case, for current to flow, both systems must be ground referenced somewhere in order to form a loop.

Is there possibly a bucking/boosting effect that can take place?

No, you would need an absurdly large reactance to achieve bucking or boosting at line frequencies.  Transformers change voltage by modifying turns ratios of common core inductors rather than with a boosting or bucking effect.  You can also take multiple transformers and parallel the inputs and series the outputs to increase voltage, but in the end this just amounts to messing with turns ratio.  If you just mean is it possible to end up with a lower or higher voltage than you expect at some point in the system, or between two points, this particular transformer connection is not particularly concerning.  On the primary it is no more dangerous than the system it's connected to and on the secondary, with a split phase output, the highest voltage to ground without transformer failure is half the line to line voltage, which is optimal for safety and significantly safer than the 460V system.

Are there dangers associated with this installation?

You're adding a less dangerous system as a load on a more dangerous system.  As long as components are correctly selected, there's nothing extra dangerous about doing that.  Beyond that, there are only the usual balancing risks to adding a single phase load to a 3 phase generator.

Am I overreacting?

Better safe than sorry.  A pound of research is worth two birds in the bush.  That said, if you don't fully understand transformers, single-phase and 3-phase systems yet, you should probably not be particularly comfortable working on them unless there is a qualified person present directing you who does.  Then again, not every Journeyman is good at 'splainin' this kind of thing and that's why electricians also go to school.