Yes, I think it is a constraint, but don't think I can tell you how different it could be from 2.73 K to cause a problem.
The CMB tells us that the universe was once much hotter ($>3000$ K), so that hydrogen nuclei and electrons had sufficient kinetic energies to remain uncombined. In the big-bang model, the universe expands and cools, then at temperatures just below this the protons
and electrons combine to form atoms that are almost transparent to the blackbody radiation that filled the universe before that. This radiation is then redshifts by the cosmological expansion and we still see it today as microwaves.
In the big-bang model, the temperature of the universe scales as $(1+z)^{-1}$, where $z$ is the redshift corresponding to a particular look back time. That we see the CMB at 2.73 K now, means it occurred at $z=1100$. Your question amounts to - would it matter if this were $z=110$ or $z=11000$?
It's less complicated to deal with the lower of these possibilities. At $z<3400$, the energy density of the universe is matter dominated, and the scale factor $a \simeq (t/t_0)^{2/3}$, where $t_0$ is the age of the universe and $a(t_0)=1$. Since $1+z = a^{-1}$, this means the connection between age and redshift is
$$ t(z) \simeq t_0 (1+z)^{-3/2}$$
If the CMB was formed at $z=1100$ this would be 370,000 years after the big bang (using the usual accepted cosmological parameters and an age of the universe of 13.7 billion years$^1$). If instead we use $z=110$, the CMB formed about 14 million years after the big bang.
That in itself is not a problem. We are just changing the expansion rate of the universe to match the redshift of the CMB and demanding this occurs just below 3000 K.
But this alters the temperature versus time relationship at all later and earlier times. How might this affect some other cosmological observations?
I think one area where the change in timescales would be hugely important is in primordial nucleosynthesis. By slowing down (or speeding up) the expansion rate, one allows an order of magnitude more (or less) time for nuclear reactions to take place.
An example: The neutron to proton ratio in the early universe freezes out at 1/6, but neutron decay with a half-life of 15 minutes, reduces the final ratio to 1/7 during the window of time available to nuclear reactions. If this window is made much bigger, the ratio will decrease further, if it is made much smaller the ratio will remain at 1/6.
Almost all of those neutrons end up in helium nuclei, so the primordial He abundance would vary between 10 H atoms per He atom through to something considerably bigger. The primordial He abundance is known to better than 1%, and it is around 12 H atoms per He atom. i.e. There was just enough time when primordial nuclear reactions were taking place for some of the free neutrons to decay, but not all. This would certainly seem to rule out order of magnitude changes in the CMB temperature.
I may add to this as other things occur to me. There is likely another constraint based on the observed formation redshift of the first stars and the epoch of reionization.