This is one of those balance issues, like weighing hiddur, rov ahm, zerizin, and bo yoser mishlucho.
Here was our response.
1. The Magen Avraham in 432:5 (as emphasized by the Pri Megadim there, but I don't get his raya from Pesachim 4b) says that the concept that it is better for one to do a mitzva personally rather than through a shaliach only requires that the person do part of the mitzva himself. Once he's done part of the mitzva, there is absolutely nothing lost by having a shaliach do the rest. The truth is, it would seem that we don't need the Magen Avraham for this, because it's obvious in the Gemara in Kiddushin 41a by the preparations for Shabbos. He didn't mop the floor, he just threw a bone into the cholent. But the truth is that the chiddush of the Magen Avraham is that I would have thought that he limited the actions he did lekavod Shabbos to little things because if he had to prepare the whole Shabbos every week, Rav wouldn't be The Great Amora Rav, he would be The Great Balabosteh Rav, so he balanced the need of mitzva bo against the time constraints. The chiddush of the Magen Avraham is that it was not a balance. Once he did a little bit, he has the entire benefit of "bo," even if his shaliach does all the rest.
This applies here in two ways.
A. Since he did the bedika himself, that is called the beginning of the biyur that takes place the next day, then according to the Magen Avraham, since he did part of the mitzva, there's nothing wrong with having a shaliach do the rest. So he should leave everything for the shaliach, and he doesn't have to do anything by himself.
B. Even assuming that the Magen Avraham would not apply where all he did was bedika, but here he is doing part of the actual destroying by himself. So he should take some with him and destroy it, and leave the rest for his shaliach to burn. This way, he did part by himself and he can do the rest via shaliach. Since the bulk of the biyur is with burning, the mitzva is being done with hiddur and he is not missing out on the "bo."
But in truth, the Chayei Adam addresses this question in 68:7.
מצוה בו יותר מבשלוחו (צ"ע אם הוא יעשה המצוה לא יהיה מהודר כ"כ כגון הרוצה לכתוב לו ס"ת ואין מכתבו מהודר ואם ישכור סופר יהיה מהודר איזה עדיףHe discusses whether it is better to write a sefer Torah yourself with less hiddur or have a sofer, qua shaliach, write it with hiddur for you. He remains uncertain le'halacha. Therefore, in our case, he would also be unsure.
It troubles me that he does not apply the Magen Avraham to kesivas sefer Torah. Evidently he held that every letter requires hiddur, so doing one letter imperfectly still forces the hiddur/shlucho issue- true, he could fulfill the "do it yourself" mandate by writing one letter, but maybe he should forego the benefit of "yourself" in favor of hiddur. We would have said that one letter without hiddur does not take away from the hiddur of the mitzva as a whole, just as we said by biyur chametz.
Another problem we have with the Chayei Adam is that we believe that the resolution to his question is clear. Rashi there in Kiddushin says that the benefit of "bo" is that he gets "tfei schar," his reward is greater. One should ask, since when is the schar a factor in deciding how to do a mitzva? One should answer that Rashi means this as a sign that it is a better mitzva. More schar proves that it is a better mitzva, and one should endeavor that mitzvos should be fulfilled in the best way. If so, in our case where it's bo but no hiddur or shlucho with hiddur, then obviously the better kiyum is with the shaliach and the bo would be serving yourself at the expense of the proper kiyum of the mitzva. We ought to do mitzvos so they are fulfilled beautifully, rather than fulfilling ourselves by doing mitzvos.
Furthermore, even if we were to assume that the "advantage" of the hiddur of sreifa and the hiddur of "bo" are commensurate, it is obvious that a hiddur in the cheftza of the mitzva, and intrinsic hiddur, is more important than a hiddur in the person doing it, which is an extrinsic hiddur.
Reb Akiva Eiger also talks about this in the gilyon to OC 446. The Magen Avraham there in sk 2 talks about finding chametz on Pesach and telling a goy to burn it, as a shvus with a mitzvah. RAE says that if you ask a goy to burn your chametz, you're not mekayeim Tashbisu because there's no shlichus l'akum.. Rav Tzvi Pesach Frank in Har Tzvi OC 143 answers that the chiyuv is to see that it's done, not a mitzva she'b'gufo, it doesn't need shlichus at all- and this is like tevillas keilim or building a maakah as discussed in the Minchas Chinuch. Additionally, some say, like the Nesivos in 182 sk 1, that ein shlichus l'akum is only where you need a chalos, but where you just need a maaseh done, yeish shlichus. Obviously, and from other rayos as well, RAE doesn't hold like the Nesivos. I think Reb Meir Simcha also says like the Nesivos, but I don't remember where.