Tevillah Bizmana Mitzvah

Rambam in Shevisas Asor 3:2 -

וְכָל חַיָּבֵי טְבִילוֹת טוֹבְלִין כְּדַרְכָּן, בֵּין בְּתִשְׁעָה בְּאָב בֵּין בְּיוֹם הַכִּפּוּרִים

According to the simple explanation of this halacha the Rambam is paskening here that tevillah bizmana mitzvah. This would mean that someone who needs to go to the mikvah (baal keri, zav, etc.) it is a mitzvah to go at the first opportunity. Thus, if that falls out on Yom Kippur or Tisha B'Av, even though there is usually a prohibition on washing one's body, the mitzvah is docheh the issur and thus the Rambam says one can go to the mikvah.

However, this Rambam stands against another Rambam which seems to hold that tevillah bizmana lav mitzvah:

Yesodai HaTorah 6:6 -

נִזְדַּמְּנָה לוֹ טְבִילָה שֶׁלְּמִצְוָה--כּוֹרֵךְ עָלָיו גֳּמִי, וְטוֹבֵל; וְאִם לֹא מָצָא גֳּמִי, מְסַבֵּב אַחֲרָיו. וְלֹא יְהַדַּק, כְּדֵי שֶׁלֹּא יָחֹץ--שֶׁלֹּא אָמְרוּ לִכְרֹךְ עָלָיו, אֵלָא מִפְּנֵי שֶׁאָסוּר לַעֲמֹד בִּפְנֵי הַשֵּׁם עָרֹם

Here the Rambam seems to hold that if the Shem Hashem is written on one's body he should even push off the tevillah in order to cover it up before he does the tevillah.

The source for both these Rambam's in the gemara in Shabbos 121a. If you look at Tosafos there (d"h Hachi) Tosafos points out a girsa issue. Tosafos prefer the girsa that even if one holds tevillah bizmana is not a mitzvah, still it is ok to be toveil on Yom Kippur or Tisha B'Av for people chayav tevillah. This would seem to be the position of the Rambam as well. Even though tevillah bizmana lav mitzvah, still one can be toveil on Yom Kippur or Tisha B'Av. The question is, why?
(See the Lechem Mishna in Yesodei HaTorah that points to this Tosafos as a possible resolution for the Rambam.)

If you look at the Rambam in Shevisas Asor in more detail it seems the Rambam himself answers the question:

ב מִי שֶׁהָיָה מְלֻכְלָךְ בְּצוֹאָה אוֹ טִיט--רוֹחֵץ מְקוֹם הַטִּנּוֹפוֹת כְּדַרְכּוֹ, וְאֵינוּ חוֹשֵׁשׁ. וּמְדִיחָה אִשָּׁה יָדָהּ אַחַת בַּמַּיִם, וְנוֹתֶנֶת פַּת לִבְנָהּ. וְהַחוֹלֶה רוֹחֵץ כְּדַרְכּוֹ, אַף עַל פִּי שְׁאֵינוּ מְסֻכָּן. וְכָל חַיָּבֵי טְבִילוֹת טוֹבְלִין כְּדַרְכָּן, בֵּין בְּתִשְׁעָה בְּאָב בֵּין בְּיוֹם הַכִּפּוּרִים

The Rambam here bunches the halacha of chayvei tevillos with someone who needs to wash dirt off his skin or one who needs to bathe because he is sick. All of these cases of rechitza are muttar. Why? Apparently it is because only rechitza of taanug, only pleasurable bathing, was part of the issur. The Rambam is saying that rechitza of chayvei tevillos was never included in the issur of rechitza in the first place.

Thus, we have two ways of learning the heter for chayvei tevillos. One way is to assume that the mitzvah of tevillah is docheh the issur rechitza. This assumes tevillah bizmana mitzvah. The other approach, that of the Rambam, assumes that rechitza of chayvei tevillos is a different type of rechitza. This type of rechitza never was assured on Yom Kippur and Tisha B'Av in the first place.