DI water at equilibrium with the air will, theoretically, have a pH between 5.7 and 5.6 for CO2 partial pressures in the range 0.0003 to 0.0004 atm. It's more likely that you would see something in the low 6's and indeed that is what you say you typically do observe. Thus 4.4 is completely unreasonable. It's worth mentioning that CO2 can pass through RO membranes to some extent so if you have source water that is alkaline with low pH...It's also worth mentyioning that it is very difficulty to measure the pH of nearly pure water as its resistivity is so high. To do it properly one uses special buffers which include salts to increase the conductivity without shifting the pH much (the small pH shift is compensated for in formulation of the buffer). The same salts are then added to low conductivity samples and the pH can be measured. STOP PRESS. That's the way Hach used to advocate doing it. They now have a new electrode especially for pH measurement in low conductivity solutions which works by virtue of large junction area. They don't seem to sell the special buffers any more.
But I don't think that's the problem. The question was "What effect does a distilled water pH that low have on the Grist/DI pH?" First let's address the question "What effect does a DI water pH that low have on the Grist/DI pH?" The answer is none. But you don't have DI water, you have distilled water. The effect of distilled water on mash pH is the same as with any other water. The pH realized when malt is mashed with this water depends on the titration curves of the water and the malt. In simpler terms the end result will depend on the relative alkalinities of the malt and the acidity of the water (for this water, assuming it's pH to really be 4.4, is quite acidic). The acidity of DI water is 0. The acidity of DI water exposed to atmospheric CO2 is tiny. In fact it isn't even acidic. The alkalinity of pure water is about 2.5 ppm as CaCO3 and that of water exposed to atmospheric CO2 about 1.8. In other words, the acidity/alkalinity of DI water used to make these test mashes should not be a factor.
Now the DI pH of Weyernanns Pils malts is between 5.6 and 5.8. You observed 5.3, a shift of 0.3 - 0.5 relative to what is reasonable. The first buffering coefficient for these malts is between 30 and 40 mEq/kg•pH. Let's go in the middle and say the pH shift is 0.4 and the buffering 35. The water would have had to supply 0.4*35 = 14 mEq/kg acid. If the ratio was 2L/kg (about a quart per pound) then the acid supplied by each liter would have had to have been 7 mEq/L. That makes your distilled water equivalent to DI water to which someone has added 0.7 mL of 88% lactic acid (roughly 10 N) per Liter.
There appear to be two possibilities here. Someone rinsed your still with strong acid and didn't rinse it or your pH electrode is shot. Which do you think the more likely? I would run the stability check on that electrode before going any further with it.
This question can't be answered as DI water doesn't have a pH that low. It has a pH of 7 (at 25 °C). So what Weyermann floor pils should have a DI mash pH around 5.6. 5.3 is way out of whack. T