For parshat Zachor, many have a practice of saying both timche et zeicher amalek and timche et zecher amalek (different vowels on זכר). What are the customs regarding this word when it appears in the regular weekly Torah reading (Beshalach or Ki Teitzei)?
The only source that exists for the Zeicher/Zecher reading is the Mishna Berura, as discussed here.
He only mentions this stringency for parshat Zechor, and not for parshat Ki Teitzei.
As to your assertion that
many have a minhag of saying both - I'm not convinced that it's accurate. For a start Sefardim and Yekkes have not adpated this Safek.
BTW: Rabbi M. M. Poliakof זצ"ל in his Sefer Minhagei Lita claims that adopting this stringency breaks one's chain of Mesora and introduces uncertainty into it. For a lively discussion in this concept, see the comments to this question.
The Mishna Berura (685 sk 18) only recommended this practice for Parashat Zakhor, seemingly because it may be a biblical obligation (ShA OC 685:7). Betzel HaChokhma (6:50) said this applies to any readings being used to fulfill the Mitzva (eg. on Purim morning, if someone missed Zakhor).
Ketzot haShulchan (3:84:13 footote 22) said to read one way in Shvi'i and one way in Maftir for Ki Teitzei and Beshallach, but to repeat the phrase for Zakhor and Purim (when there is no second chance). R Seraya Devlitzky (Eim leMikra) ruled this way, but in the case of Zakhor prefers repeating the whole verse. R MM Schneerson wrote that since Maftir is not on the same level as Shvi'i, one should repeat the phrase in each. It appears (Nefesh HaRav p111, MiPeninei HaRav 190) that R JB Soloveitchik also repeated the word in every instance, though I've heard rumors that he repeated the whole verse every time.
R Moshe Feinstein wrote (Igrot Moshe OC 5:20:32) that there is no need to repeat the word qua Parshat Zakhor (as there is no basis for requiring perfect pronunciation to fulfill the Mitzva of remembering Amalek (cf. Benei Vanim 2:1)) but rather qua Torah reading, and so it should be done at any Torah reading.
R Moshe Shternbuch (Moadim uZmannim 2:167, quoting R Meshullam Roth (Besoret Eliyahu 5)) felt that only for readings from Ki Teitzei is there reason to repeat because there it is the command to us which we must not misinterpret, as Yoav did (BB 21b).
This is of course all of only historical interest nowadays when we know the authentic Jewish tradition is to read Zeikher with 5 dots.
Rabbi Moshe Feinstein writes in the Igros Moshe Orach Chaim vol 5 siman 20 subsection 32 that it is more important to repeat it when reading it in parshas Ki Seitzei than in parshas Zachor.
But, just for the record, the Vilna Gaon also repeated it during Ashrei, and no one seems to have accepted that practice.
I've been in one shul where they have found a compromise, if you can call it that. The shul wasn't as much concerned about the pronunciation issue of "zecher" as much as it was concerned about questions of repeating verses either in whole or in part. (There are various opinions regarding this, and different people in the congregation would "argue" about this aspect.)
Realizing that the entire 3 verses of zachor Amalek are read twice - once as the end of shvi'i and again for maftir, they pronounced it as "zeicher" for shvi'i and "zecher" for maftir.
I am not sure why the confusion arose in the first place. If I recall the Mishnah Brurah, he also doesn't discuss why the minhag originated. The correct Hebrew dikduk should be "zeicher" with a tzeirei, as the word is an adjective. As a general rule, segol becomes tzeirei when used as an adjective before a noun, such as the word "mateh" which becomes "Matei" as in "Matei Levi".
(Aside, I am uncertain why in Kiddush it is pronounced "zeicher l'ma'aseh breishit" unless "zeicher" is considered an adjective in this phrase?)
As discussed here the Lubavitcher Rebbe equates both readings - if you are reading it one way on one occasion, you should be consistent on all occasions. This is the Chabad custom.