The Rashba[1] posits that the laws of chatzitza do not apply to the tefillin shel rosh. However, the accepted opinion is that a chatzitza is problematic even for the shel rosh, although possibly only regarding the bayit but not the retzuot (straps).[2]
Many poskim[3] posit that a removable toupee is a chatzitza. However, Rav Moshe Feinstein[4] maintains that transplanted hair is not a chatzitza, since it becomes a permanent, desired part of one’s body. Furthermore, he writes that this is also true regarding a permanently glued-on toupee.
We must still ask whether a toupee that is glued down for only a matter of months is considered a temporary or a permanent appendage to the body. The status of appendages that remain for an extended period are discussed prominently regarding items such as removable stitches and temporary fillings for women immersing in the mikveh. In that context, many poskim[5] are lenient and permit tevilla. One of the lenient factors[6] is that the ostensible chatzitza is something that is specifically needed for medical reasons for a significant amount of time. This factor is absent in your case. However, several poskim are also lenient in a case of aesthetic need, e.g., to enable a married woman to have braces on her teeth,[7] and such a need is parallel to your case. Your case is actually better than that of braces, since people want to remove the braces as soon as possible, whereas you would presumably want to keep the toupee on as long as you can.
There are various opinions regarding how long the item must remain on the body in order for it to be considered permanent enough that it is not considered a chatzitza: six months, one month, or a week.[8] If, for example, the halachically required time for permanence is a month, then according to some opinions, the appendage becomes a chatzitza a month before it will be removed; others say that once it is on for a month, it does not constitute a chatzitza as long as it is there.[9] Your situation is easier to permit if the toupee will be removed in order to be re-glued rather than to be replaced, as it will essentially continue to stay on your head after what one can argue is an insignificant pause.[10] Moreover, a woman who wants to follow the stringent opinion can synchronize going to the mikveh and removing the appendage; in contrast, a man who has to put on tefillin every day cannot.
Let us halachically contrast tevilla and tefillin. On the one hand, tevilla is necessary in the context of a more stringent halachic matter than tefillin. Furthermore, as noted above, there is an opinion that chatzitza is not a problem for the tefillin shel rosh at all. In other respects, however, your case is more severe than most questions regarding tevilla. For one, a chatzitza on a minority of the body during tevilla (as in the cases above) is no worse than a Rabbinic disqualification.[11] In contrast, in your case, the entire area of the tefillin shel rosh is covered by a toupee, and there is thus the potential for a Torah-level disqualification. Some authorities argue that the parameters of chatzitza for tefillin are stricter than those for tevilla,[12] and some actually claim that even one’s own hair that is under the tefillin in an unnatural way is a chatzitza.[13] A toupee should be no better than such hair.
In summary, it is likely that a toupee that is glued down in a semi-permanent manner would not constitute a chatzitza as long as it is still considered desirable. This would allow one to put on the tefillin with a beracha. However, it is clear that according to significant opinions, the mitzva of tefillin could be compromised. A removable toupee, while more clearly a chatzitza, has a major advantage in that a solution is possible. Several poskim suggest that one who is embarrassed to remove the toupee publicly put on tefillin at home without the toupee, recite the beracha and say Kri’at Shema there, and then daven in shul with tefillin on the toupee, without reciting a beracha a second time.[14]