Why does my 12V SLA Battery behave strangely when charged by simple LM317 circuit?
I want to post an answer to my question having learned a lesson or two and to clear up some observations I put in my question that were incorrect.

Firstly, subsequent tests using the power supply showed the same behaviour as the LM317 charging circuit.  The voltage starts low and quickly rises to the set constant voltage value.  Current limiting is activated at the start, but the current quickly reduces (the battery I'm using is not deeply discharged).  I'm not sure why I saw different behaviour with the P.S, but what I saw shouldn't have happened (or I was seeing things!).

In the end, I built a modified version of the LM317 circuit that operates in 3 different modes.  It has current limiting for the starting phase of charging, when (as I've observed) the battery will accept a large amount of current.  As the current flowing into the battery falls (and it will as the battery charges) the charger voltage will rise and will be limited to 14.7V.  The battery will remain charging at a constant 14.7V until a lower current threshold is passed (around 120mA), at which point the charger flips into a float charge mode and keeps the battery charged indefinitely at 13.7V.  The current flowing into the battery reduces all the while and eventually very little current is flowing.

The XH-M603 charge controller did not work for me.  This module accepts a user-defined lower and upper voltage level.  If the voltage on the battery (during charging) is lower than the lower limit, the relay closes and the battery charger (the LM317 circuit in my case) charges the battery.  If the voltage on the battery (during charging) is higher than the high limit, the relay opens and the charger is disconnected.

The problem is, the voltage on the battery terminals during charging is always the chargers output voltage (or a value that's close).  If I set the upper limit at 13.7V (the voltage that ideally I'd like the battery charged to), then this voltage is easily exceeded during the constant voltage charge phase and the controller simply switches off.  With the controller off, the battery terminal voltage settles back to its State of Charge voltage and lets say I'd set the minimum controller voltage to 13.6V (because I don't want the battery discharged below this voltage), then the controller switches back on again.

So, the controller cycles constantly on and off (until the relay wears out).

If I set the upper limit at 14.7V or higher, then the constant voltage operation of the charger won't allow the charger to exceed this value and the controller will never switch off.  So, basically this module is useless for my use case.