When I was using tap water direct into my system, it would mean I constantly had to add PH down to keep the PH where I wanted, even with roots lowering PH when feeding. I found that, like you, I had a constantly rising PH no matter what I did.
Then I got an RO system and I find that my PH will wander up and down, with no particular direction until the plants start to feed heavily at which point it will constantly lower, meaning I have to add PH up to get the PH I want. This normally happens by about week 2 of flower. Until that point, if I want a particular PH I need to add PH up AND PH down to buffer the solution and allow it to be stable.
It is because of issues like this that I now use an automated dosing system. it looks at your nute levels, and PH and if they go out of range, adjusts them to be in range. It does this every 10 minutes, even while you sleep. When I refill my system I just turn on the valve to the RO system, it starts adding RO water to the tank, when the level is high enough I start the pumps and the doser which then puts nute levels and PH within range before the tank is full and keeps both within range as it fills and from then on. When the tank EC and PH are within range I add plants.
As such, I can leave my plants for weeks at a time, as long as I can see my graphs I know what they are doing.
In tap water there are usually high levels of calcium etc which will buffer the water to a high PH meaning to maintain a set, lower, PH you need to constantly add PH down. As water is taken up or evaporates, more high PH water is added meaning that, again, you have to add PH down. This is fairly simple, although frustrating.
If you get an RO system, you will find you need to be even more vigilant on your PH as it will vary in two directions instead of only one. It will even possibly vary more as the water is unbuffered.
It seems that your lava rock, your perlite, your tap water AND your nutes all raise PH. All that getting RO will achieve is meaning your lava rock, your perlite and your nutes are raising PH.
Personally, I would fit a float valve running to a tank filled with water slightly more acidic than you require, then when your plants take up water, or water evaporates, the replacement water will slightly compensate for the raising PH.
Getting RO will make your system more volatile, which means more growth if you have everything dialled in, more headaches if you have not.