Answer

At first glance, no, because this is a doubtful positive commandment against a doubtful prohibition (the doubt is whether he is applying the blood that requires four applications or the blood that requires one). But on the simple level, when liquid mixes with liquid, the mixing is complete, and therefore each application contains both types of blood. If so, you are seemingly right that this is truly a case of a positive commandment overriding a prohibition.
But one can distinguish between them, because the positive commandment is fulfilled through one type of blood, while the prohibition applies through the other type of blood. In such a case, one could say that this is a commandment fulfilled through a transgression, not a positive commandment overriding a prohibition (as is well known from the medieval authorities, such as Rabbi David in Pesachim, and many others). And perhaps this itself is the point of dispute between the tannaim in that Talmudic passage. But from the reason you gave for Rabbi Yehoshua—that he bases it on the fact that “do not subtract” is violated through passive omission, and not on the issue of a positive commandment overriding a prohibition—it sounds as though even in his view the positive commandment does not override the prohibition, except that because the transgression is by passive omission, that is preferable.
There is more room for analysis here, because if he does this out of doubt in order to fulfill a commandment, perhaps there is no violation of “do not add” or “do not subtract” here at all. Even though ordinarily no specific intention to add or subtract is required, it seems that here it is hard to define it as “do not add,” because he is acting out of a desire to fulfill a commandment (that is, he intends to fulfill a commandment, not merely lacks intent to add).
Indeed, this is a topic worthy of a great deal of analysis, and I wrote what occurred to me just now, off the cuff and not as a settled conclusion. More power to you.