The nickname for what you are doing is called "HV Reset", and it does seem to be the best way to reduce engine noise during a climb. As for reducing stress on the engine, that assumes that there is stress worth worrying about. High revs and the associated loud noise generally bother us more than they bother the engine.
I agree there should be an option for handling this better. Hybrids most of the time run in one of two modes, charge depleting mode (aka EV mode) and charge sustaining mode (aka HV mode). There is another mode at least at the terminology level called Blended Mode where in a high load situation the system will essentially combine charge deplete with ICE usage, using the engine for power to assist the battery in a high load situation rather than trying to maintain battery level. Most hybrids including Clarity have the ability to simultaneously power the wheels with both battery and ICE, however this usually only happens during brief periods of acceleration, and once the SOC starts dropping the system prioritizes ICE for battery charging rather than providing power to the wheels. Or else the engine provides both power to the wheels and battery charging at the same time, resulting in high revs and noise. Blended Mode would be a mode that you can activate for prolonged periods like in a hill climb, where the priority for ICE is to assist the battery in providing power when there is a medium to large power demand.
The only PHEV that I know of that may do this is the Chevy Volt which has something called Mountain Mode, but the way I read it that may just be a more specialized version of HV Mode (known as Hold Mode on the Volt). Hold Mode, like our HV Mode, sustains the battery at whatever SOC it was at when you switched to Hold Mode. Whereas Mountain Mode targets 45% SOC. The idea being that if you know that at some point on your trip you will have a hill climb, you can engage Mountain Mode at the start of the trip then you don't have to think about it as it will burn down the EV charge until it gets to 45% then essentially go into Hold Mode and maintain SOC at that level. The idea being that 45% charge should be enough to make it up most hill climbs with just the battery. Apparently when it later senses a prolonged load it ignores the 45% target allowing the battery to start depleting, but if I understand correctly it's not in blended mode, it just essentially reverts back to EV mode for the hill climb. If I understand correctly on a long hill climb even with Mountain Mode active the Volt can run into the same problem as Clarity, once the battery is depleted then you have to make it up the rest of the way solely on ICE. I don't know this for a fact I am just basing this on something I read from someone who drives a Volt in Kona, Hawaii, they say that for the big hill climbs at high speed they use Hold mode at the start of the trip so that they can start the climb with a full battery. They said this works must better than Mountain Mode. Making me think like I said that Mountain Mode is just a specialized version of HV Mode, it is not a Blended Mode which is what is really needed.
But mountain climbing is a niche requirement, in a type of car that is already very niche (PHEV), so this is just one of many refinements that are lacking in PHEV's that probably won't be addressed unless or until PHEV's become more popular. Actually Blended Mode would be nice even around town, as it would keep the engine quieter while accelerating from a stop light, by using ICE at a lower RPM and being willing to sacrifice some SOC. Then when you reach cruising speed ICE can slightly raise the RPM to recharge the battery prior to the next stop, which you likely won't even notice since you will have more road and wind noise at 45 mph or whatever. Especially if in blended mode the system will be more relaxed about maintaining SOC, allowing SOC to drop if necessary to maintain lower RPM. It wouldn't work perfectly all the time but it would certainly be better than the current situation, where if the SOC is just a hair below the target SOC when the light turns green, you can get an embarrassing growl from the engine, making your passengers wonder if something is wrong with your car.