I am going to piece together a variety of bits of information, so please bear with me.
- Eliezer was in charge of Avraham's house already before Avraham had his name changed (Bereishit 15:2)
- Eliezer was an "eved" of Avraham (Bereishit 24 - there seems to be agreement that the eved not named refers to Eliezer).
- Avraham circumcised all the males of his household, including non Jewish slaves (17:13, 23) which would include Eliezer.
- A non-Jewish slave even not in Avraham's time is subject to some form of conversion during his servitude and must commit to the process within 12 months after being purchased or must be sold (from here)
- Eliezer's conversion was completed - he remained with Avraham for more than 12 months(Radak to 15:2 who states that Avraham actually gave the servant "Damesek" the new Ivri name of "Eliezer" which would point to a completed conversion, and would point to the idea that the avot followed the torah)
If these are the case, by the time Avraham sends Eliezer to find a wife for his son, he should not have resisted the possibility that Yitzchak could marry Eliezer's daughter. At that point, though Eliezer might once have been included in the curse on K'na'an (as discussed here), as a convert, Eliezer should have been viewed as a complete Jew and not called out in his status as convert (as discussed here). So why does Avraham reject him and remind him of his ancestry as a K'na'ani?