With PLAY v1.6 there might be some differences, but the new and improved power design from PLAY v2.2, PLAYMATE and SWING will, most likely, not be influenced by the input power supply. Worth mentioning that included SMPS power brick from BURSON is really low noise, so spending money on an external power supply might not be so worthy.
PLAY v2.2 with default SMPS
PLAY v2.2 with my DIY Linear PSU
The only difference is on the 3rd harmonic that decreases from -102dB to about -120dB (which is a, good thing actually), but the THD is barely improved. Of course, this improved will not be audible in any way.
Around 400uV of AC RMS & noise on my DIY Linear PSU on 12V/3A resistive load. PLAY/PLAYMATE/SWING will only take about 0.6A when playing, so LPSU's noise will even be lower, barely measurable by my scope.
I personally
do not recommend swapping the default SMPS power brick provided by BURSON, my main concern being related to
loss of the warranty, combined with
zero audible improvements (although, slight improvements might appear on dedicated measurements).
Please, do understand that inside PLAY/PLAYMATE/SWING/FUN there are dedicated power regulators that care little to none about the main power supply used! Worth telling that these regulators might not like so much the Linear PSU used by yourselves and the possibility of oscillations are raising some concerns (it happened to me with FUN where inside regulators didn't liked my DIY linear power supply).
L.E.: Actually, Alex from BURSON told me that swapping the power brick will not void the warranty, so feel free to knock yourself out, just not expect any miracles.