This entire subject opens a major can of worms and leads you to rabbit holes you might fall into. And stepping outside of the modern line of reasoning here is not a great way to win friends (ask me how I know). Despite taking much flak from moderns who follow myriads of modern era magazines and books (typically not peer reviewed) of pablum/mantra published with an opinion to the contrary, I've come to the opinion that much of the older peer reviewed literature indicates upon a careful analysis that mash pH's were (back then) measured at mash temperature. Sometimes this is even made explicit and comparisons to room temperature are made (almost universally by adding 0.3).
This implies that mash pH back then was something accomplished at a range somewhat higher than is considered normal today, wherein we tend to look at all pH related things in terms of their room temperature pH measure, and we also somehow collectively contort the masters words of yore to conform to our modern worldview that pH is always to be taken at room temperature, thus seeing their 5.4 ideal as being at room temperature (albeit that it was not).
Off the top of my head it was Briggs who most explicitly expressed this, when he openly stated in writing that his ideal mash pH range when converted to room temperature (wherein he specifies an added 0.3 pH points to facilitate the conversion) becomes 5.5 to 5.8 pH. Similar data can be teased from others such as Narziss and the rest if one desires to take the effort to do so. Modern pH instrumentation seems to more closely adhere to a difference of 0.2 points when converting from mash temperature to room temperature pH readings, so for today's pH meters Briggs room temperature ideal mash pH range becomes 5.4 to 5.7 (with a mid-range ideal of 5.55).
But there is a catch. Today we tend to conflate what was once two separate pH adjustments that were commonly made in yore into only one. And we generally presume that if we mash at ~5.4 as measured at room temperature we will exit the boil at a cooled to room temperature Wort pH of ~5.1-5.2. The masters of yore acid adjusted the boil to 5.2 as measured at room temperature (after earlier having adjusted the (room temperature) "Mash" to 5.5 - 5.8 pH (5.4 to 5.7 for a modern meter), thus making two adjustments.
All literature mentions a drop in pH across the boil, but some of the peer reviewed literature which actually took the time to measure this (as opposed to regurgitating the presumptions of others) speaks of a drop in pH across the boil that is greater in magnitude when entering the boil with Wort at room temperature pH's of 5.4 and above, and with the observation that once Wort is at ~5.2 pH (room temperature) or below this pH drop witnessed across the boil ceases. So the magnitude of the pH drop across the boil is greater the higher ones pH is going into the boil, but somewhere around pH 5.1-5.2 any further drop across the boil ceases. The boil drop is therefore not fixed in magnitude (typically with a presumed 0.3 point drop), as is commonly believed today, and was presumed incorrectly even as such within much peer reviewed literature of yore.
I believe it was Bamforth who made it clear that this pH drop across the boil can not be relied upon to occur at any predictable degree of reliable measure (or magnitude), so the presumption that we make today of hitting 5.2 post boil and cooling when starting at around 5.4 at room temperature in the mash is not something we should "take to the bank" (such as we do).
But since once a room temperature measured ~5.2 pH is achieved there is little to no further drop in pH anticipated to be observed across the boil, your choice of mashing at room temperature 5.2 and then lautering, sparging, and boiling should easily accomplish the task of exiting the boil at 5.2 pH or right close (perhaps as low as 5.1). 5.2 mash pH may however be below the ideals seen for enzymes and saccharification yield and the like, as 5.55 room temperature pH for the mash seems to best tickle all of these buttons (as attested by the masters of yore). By considering mashing at room temperature 5.2 pH you are among those (nearly all of us) today who have conflated two acidification steps into one. But a growing trend is emerging to once again revert to two pH adjustments.