My experience: by the time you purchase a steel barrel Such as a Bartlein and then have it fluted, it is not that much of a savings over a CF barrel. Just saying.I went through this exercise...twice in the last 2 years and about to be a 3rd and 4th time this year as I'm building a switch barrel on a short action. I went with Proof on the last 2, as volume won't be high (they're for hunting, not high volume steel or comp). Weight was a factor on these builds.
The switch barrel build I'm currently considering will be a Proof for one barrel (weight is the consideration) because it will be easy on barrel life so even if I do some plinking it won't kill my investment in a year. The other barrel will likely be a fluted #3 from Bartlein. Again, weight is the consideration here but also barrel life- this chambering will be more of a barrel burner and so I don't want to shell out ~$700 for a barrel plus smithing fees to burn it out quickly. I'll sacrifice a little weight here, but I can buy 2 of these Bartlein barrels for the cost of a single Proof, and given that the barrel life will be less than half of the others I've done in different chamberings, that seems like the better thing to do FOR ME.
Regarding use of the Proof- at the range I haven't had a problem with heat. I live in TX though, so I'm always overly cautious. Out of habit, for carbon or steel, I don't shoot any more than 5 rounds in a string, with at least 30-60 seconds in between trigger pulls. After the 5th round, I set the rifle aside and grab another to work with. I try to let each barrel cool for 10 minutes or so, maybe more depending on outside temp. The exception here are some very heavy steel barrels I have that were purpose built for 10 round strings...still, I then let cool and switch to another rifle, take breaks, shoot the $h!t with other range patrons, etc.
Like @FEENIX said, though, personal preference.