I've used a BEC and it works fine (even cheap ones will do over 2A easily). To clean up any possible dirty voltage, add a capacitor to the output - 1000uF or higher rated for 10v or more will be fine. Low esr is better but not required. A similar input cap, but rated for ~20v, wouldn't hurt either. You can even add a ferrite donut to the output wires.
However, the small contacts of a micro usb connector tend to drop a little voltage as charge current increases, so the phone is likely only getting 4.8-4.9v when trying to charge at higher rates. The included charger probably boosts the 5v up a little to compensate. So I would bump up the BEC output so the phone sees a solid 5v. Usually 5.1v-5.15v is good and is still well within the USB spec (spec allows up to 5.25v). The method to boost voltage varies by BEC maker. For a CC BEC, it's programmable - just be sure to
measure the output (I've seen actual measured output vary considerably from the programmed setting). For other BEC types, I've
modded some in the past to provide a specific voltage; it's not too hard as long as your soldering skills are adequate.
Shorting the two data lines is
typically what you'd do to enable high speed charging, but some phones/tablets use other methods. Some older Samsung phones/tablets (galaxy s3 era) need a 200 ohm resistor between the D+ and D- lines instead of shorting. Apple uses a completely different scheme involving voltage dividers between the power and ground being fed to the data lines. And, yes, some even use that sense line - although it seems to be fairly uncommon. So, before you assume shorting the lines will work, check forums for your specific phone.
Just be aware that ANY bec can fail where it will send full input voltage to output (in your case: ~14.4v). Any decent USB charger will contain a little extra protection circuitry, but this is missing in any BEC I've seen. The chances of a BEC going bad when being used at moderate levels though is small, but just thought I'd state that warning seeing as how it will be hooked to a device costing a few hundred dollars.