Why can you put the Chagiga on the table with the Pesach?

Question

The end of Pesachim 9:8 says:

נִתְעָרֵב בִּבְכוֹרוֹת, רַבִּי שִׁמְעוֹן אוֹמֵר, אִם חֲבוּרַת כֹּהֲנִים, יֹאכֵלוּ: ‏

If the Pesach got mixed up with Bechoros, Rabbi Shimon says, if it's a group of Kohanim they should eat it.

All of the procedures for bringing a Pesach and a Bechor are identical. However, a Bechor can be eaten for two days and a Pesach can only be eaten for one night. Rabbi Shimon allows bringing the unknown korbanos, even though the result is that the Bechoros can only be eaten for less time than ordinarily allowed. Although it's not explicit here, the Chachamim disagree with Rabbi Shimon (אין מביאין קדשים לבית הפסול), and the Rambam follows them.

However, the Rambam also rules, following the Tosefta, that

בְּשַׂר חֲגִיגָה שֶׁעָלָה עִם הַפֶּסַח עַל הַשֻּׁלְחָן וְכָל הַתַּבְשִׁילִין הָעוֹלִים עִמּוֹ עַל הַשֻּׁלְחָן מִתְבַּעֲרִין עִמּוֹ. וְאֵינָן נֶאֱכָלִים אֶלָּא עַד חֲצוֹת כַּפֶּסַח עַצְמוֹ גְּזֵרָה מִפְּנֵי הַתַּעֲרֹבֶת: ‏

Chagigah meat and other cooked foods that were on the table with the Pesach are burned with it. And they can only be eaten until midnight, like the Pesach. This is a decree because they might get mixed up.

Why are you allowed to put the Chagigah on the table with the Pesach when this limits the time it's allowed to be eaten?

Answer 1

Joel K suggested in the comments that, in fact, you're not allowed to put the Chagiga on the table with the Pesach lechatchila, for this reason. The Rambam relies on the other places where he paskens that אין מביאין קדשים לבית הפסול, such as Chagiga 2:13.

Double AA supported this with a diyuk from the Rambam's language: for the Chagiga he says שעלה, meaning it already happened, while for other, chulin dishes, he says העולים, implying that you're allowed to put them there.

The Tosefta, which is the Rambam's source, uses העולה for both. Apparently, that Tosefta is following R' Shimon. The Rambam paskens the Tosefta, but adapts it to our ruling, which is not like R' Shimon.

I haven't found anyone making any of these points in the standard commentaries on the side of the Rambam or Tosefta or in the back of the Frankel Rambam.