This doesn't seem to make much sense at all.
What are you trying to do? If you want to measure the mass gained by a body due to heating, the last thing you want is a radiating system, because you will have a devil of a job working out how much heat has been lost, while you have been putting heat in. You want something that can be thermally isolated: a calorimeter, in fact. Don't you? Why use a laser and futz about with Stefan's Law and all the problem of radiative loss? You can heat it electrically.
And if you think the system comes to equilibrium at only 27C (2C above standard "room temperature"), what temperature are you assuming the block has when you start? Have you made a typo somewhere?
No, the predicted mass gain from thermal effects is too small to measure in practice, so I think you are barking up the wrong tree with this idea. You are far better off to consider the nuclear mass defect. Now that is something that confirms E=mc² in a very obvious and indisputable way.
I know, it's wrong but I let myself enter into my own trap.
I used black bodies and lasers to have at hand the speed of light "c" in some way.
I used Stefan's law to introduce body's temperature, but it didn't work. When I used 700°K as temperature at
thermal equilibrium, it gave me that the body irradiated 11.5 Watts, while being heated with a 10 Watts laser!
Because of that I lowered the temperature. I cheated but failed anyways.
Mea culpa: I presented the problem in a hurry, without proper considerations.
Still, I can say that "in the internet" there is consensus that a heated body gain mass due to energy absorption.
I didn't explore it any further, so I started to play with simple theories.
This "feeling" that there is a relationship between mass and energy is older than written history.
Once, I imagined Newton observing how a piece of paper burn with flames and smoke, and I thought
that we might be asking himself: "Where did the matter of the paper go?. Some of it converts into
radiant heat (the concept was known by then); some is converted into convected heat; some part
is converted into smoke and there are carbonized remains. How do all of those parts relate to the whole?"
But this is just me. Don't pay attention and let the topic die.