The ineptitude of the general public regarding charging of starting batteries proves that batteries can and usually do 'just fine' with whatever sort of voltages the vehicle allows.
But we here on this forum know that battery performance and longevity can be greatly improved by proper recharging, which means charge to full, promptly after any significant discharge.
While the heat the underhood battery is exposed to is controllable only to a slight degree by the owner, its recharging cam be amended to a much superior status by the use of external charging sources and the simple knowledge that the vehicle simply does not care about battery longevity and the voltages allowed by it are far from ideal.
A newly purchased AGM reading 12.68v has obviously been sitting on the shelf for a while. Top charging it with the adjustable power supply is very wise. One deep cycle, even on a AGM battery designed for starting purposes only, is not going to damage it, provided it is recharged promptly and to full.
I was unable to get my ~1 month old Northstar AGM, when I purchased it, via a shumacher 'smart' charger to read over 12.85v. While I could tell its cranking amps were quite high as it cranked my engine with authority, it was not until i discharged it to ~50% and recharged it at 25+ amps until full, that its blisteringly high pulse cranking amps became obvious, and thereafter it rested fully charged at 13.06v, and startled me every time I twisted the key to start.
This battery is now, right around the 750 Deep cycle mark. It will have been my sole battery since June 2015 for both house loads and engine starting, it will turn 5 years old in November. The voltage it holds under load for AH removed from it, is still impressive. It has lost much of that extreme ability to crank the engine so fast that it is scary, but it still can start my engine easly depleted 65 of its original 90AH rating, and it only drops to 9.73 volts when depleted 65 of its 90 AH under the ~140 amp load of the starter.
Since you have the tools, and ability, and obvious desire to get the best performance and longevity from your new AGM battery, I would say, after following the protocol outlined by mex to top charge it) to discharge the thing to the 50% range and recharge it at 25+ amps until it reaches 14.7v and until amps taper to under 0.4. When you do turn the key to start the engine thereafter you might be shocked at just how fast it cranks and fires up, and its resting full charge voltage will likely be 12.95+ .
Or you can say it will be 'just fine' as is, and it very likely will be.
But working 'just fine' is subjective and in some instances can be little better than 'it will fail tomorrow'.
I'd rather have something work to its maximum potential, as opposed to 'just fine'.
An Adjustable voltage charging source, and an ammeter will allow you to achieve maximum potential, if you choose to do so.
Turning on the headlamps on high beam is likely a 15 amp load. 3 hours of it should take the battery to the 50% range. The goose it to 14,7v with 25+ amps, and hold it there until amps taper to 0.4 or less.
