The amount of heat transferred may be the same, but it's unlikely that the
temperature will remain constant throughout the process. Depending on environmental
conditions, the regulator (as well as the tank) will act as a conductor, dissipating
or absorbing heat, or both, and will affect the total heat at any given point in the
process.

Heat = mass * specific heat * T.  Delta heat = mass * specific heat * delta T.

If the regulator transfers enough heat to a cold environment that can cause the heat
loss that leads to freezing. As far as temperature changes and heat transfer due to
expansion, the regulator doesn't cause them because the regulator doesn't cause the
expansion.

It doesn't matter where the tank is - unless it's suspended in a
vacuum inside a mirrored chamber - its temperature will seek
equilibrium with the surrounding environment. (Giving up or acquiring
heat in the process.) The heat transfer mechanism is radiant and
convective/conductive.

Expansion of the gas through an orifice (regulator) will cause cooling
of the gas to a temperature below any ambient. If the rate of cooling
due to expansion is faster than the rate of heat replacement due to
radiant/convective/conductive processes with the ambient then the
temperature differential with respect to the ambient will be large. If
the ambient temperature is close to freezing then it will take very
little differential to freeze the regulator.

I don't know if you followed all the links related to that link I
posted originally, so I provide a new one here.

http://scienceworld.wolfram.com/physics/CharlesLaw.html

Here we see that pressure, temperature and volume are all linearly
related and we find a nice magic number: 273

Why does that number sound familiar? Hint: It's a Kelvin number.

There are three problems with application of Charles' Law here: It
applies only to ideal gasses and air is not an ideal gas, it neglects
any heat input from the environment and it neglects tensile expansion
of the container due to the pressure of the gas. Charles' law assumes
zero heat transfer and a perfectly rigid container and a monatomic
ideal gas but as a first-approximation sketch of what can happen it
can serve as a good starting point for a thought experiment.

Assume an ambient of 274 K (1 degree Celsius) and that our tank is at
ambient. We can easily calculate the volume of ideal gas we need to
remove from the initial  volume to lower that tank to 273 K and freeze
any water in it:

V = V0(1 + (-1)/273) so if V0 = 1 then V = .99633(V0) or 99.6% of the
original volume. Less than 1% of the gas in the tank needs to be
removed to effect a 1 degree (Celsius/Kelvin) change in the gas
temperature. In the case of an 80 ft^3 tank, that would be about 0.8
ft^3, right? How many breaths would it take to remove that volume of
air? If you are doing an under-ice lake dive would you be in trouble
if you had water in the regulator components?

In the real world, thermal transfer from the ambient to the container
and gas would tend to increase the quantity of gas release needed to
effect a given temperature change. Any ambient temperature that was
NOT -273.15 K would allow heat transfer into the gas. (Even freezing
water has "heat".) The effect of 274 K water around our regulator
would be to increase the rate and gas volume removal required to
achieve the desired temperature change.

Let's now assume a nice warm Southern California dive, an ambient of
68 F, or 293 K:

V = V0(1 - 20/273) = .9267 or 92.7% of original volume.

So we would have to remove approximately 10% of the original air in
the tank rapidly enough to overcome thermal transfers from the
environment in order to freeze a regulator. Of course, this assumes
there is water in the regulator to be frozen. Without liquid water
there can be no freezing. Thermal expansion coefficients of the metal
components of modern regulators are small enough that it would take a
very large temperature change to cause seizing of the mechanism.
Minimizing use of dissimilar metals in regulator design also minimizes
this risk.