Yes, it is possible. This shares the similar opinion with English counterpart, whether conjunctions can be used to start a sentence (Yes indeed, you can).
Begin with または or not
For written Japanese language, look at the full-text search result of または on ウィキペディア for example. From the first 20 matching items, I have found only one article i.e. フィード that uses
または at the beginning of a sentence (Text in bold).
フィード（英: feed）とは、ウェブサイト、特にブログやニュースサイトなどのコンテンツの概要もしくはコンテンツ全体を配信用に加工した文書のこと。またはそのファイルフォーマットのことをいう。 [...]
Even within a page of the first 100 matching items,
。または (include period when using 'Find' in web browser) had only 9 matches, which is less than 10%. Therefore, we can assume that Japanese articles often use it somewhere in the middle of sentences.
So, is it possible to begin a sentence with または? Yes. Should one begin as such? Not quite, especially when conversation could be expressed in other ways.
Notice that the use of conjunction "or" in given conversations are not intended to limit the choices of the listener to reply, but for the speaker to give suggestions in subsequent order. For this reason, the conjunction at beginning would be better removed at all.
In the first conversation, I'd suggest to remove the words
または. Then, I'd suggest to replace
...でも(even; also) with
...とか (such as; like).
When shall we meet?
How about today? May be (like) Wednesday?
In the second conversation, the words are removed similarly. Assuming "was not on the list" is not because name was not written earlier, I'd suggest to replace
書いてなかった (did not write) with
載ってなかった (not mentioned). Otherwise, it is
書かれてなかった (was not written).
My name was not mentioned/not written on the list.
You must have signed up too late. May be it was just a mistake?
If the speaker feels
かも? cause the speech to end abruptly, consider to add
しれない。 to the sentence and remove the question mark.
[...] May be it was just a mistake.
間違いだけだった of original sentence, this translates to "was [the] only mistake". In contrast,
間違いだっただけ of revised sentence translates to "was a mistake only" and English could rearrange to "was only (just) a mistake". This may be tricky to understand.
Above all, the idea is to give up the habit of using
または at the beginning of sentences. The repetitive use of these conjunctions is quite awkward to hear in conversations.