回答
以下の説明は少し把握しにくいかもしれません。お質問があれば聞いてください。
The intended meaning is adequately conveyed if you use "for", and would be accepted by English speakers, but it is slightly more correct to not use "for".
That is, it is slightly more correct to say:
"I lost the watch which my father had bought me."
than
"I lost the watch which my father had bought for me."
Consider the structure:
I lost [the watch [which [my father had bought me]]]
==
I lost [the watch [which [my father had bought me {the watch}]]]
{the watch} is deleted during the derivation according to standard theory.
Let's examine the phrase in question further:
[my father had bought me {the watch}]
Here we have a verb structure for "bought" which includes two objects. One object is direct ("the watch"), the other is indirect ("me").
In this case, the semantics (言葉の内部の意味構造のようなもの) of "bought" imply that the indirect object is the actual recipient of the item that is bought. I assume this is the intended meaning of your sentence.
On the other hand, consider the alternative you mentioned, using "for":
[my father had bought {the watch} [for me]]
This new phrase "for me" is more generic. This usage/meaning of "for [...]" can be used with many verbs, and it basically means, as you probably already know, that something is done for the benefit of someone else. Similar to してもらう／してあげる.
"She submitted the paperwork for me yesterday."
"He opened the door for me."
However, since it is generic, "for [...]" cannot contain information about whether someone is the actual recipient of the physical object in the case of "bought the watch for me."
Any implication that we derive from the use of "for [...]" is a result of the combination of "for [...]" with its surrounding structure. In particular:
my father had bought {the watch} [for me]]
We are likely to infer that I am the recipient of the watch. Implicitly, we think that there is no likely reason that my father would buy such a thing for my benefit, other than that I am its recipient. This conclusion comes from the combination of the meaning of "bought" with the generic "for [...]" structure, as well as the meaning of "the watch", to some extent.
However, there could be a completely different interpretation of "for [...]", depending on the context. Consider the following slightly absurd scenario.
Guard: "Halt! You cannot enter this shopping center!"
John: "Why not?"
Guard: "Do you see that store over there? Someone needs to buy a watch at that store. Until then, you may not enter the shopping center."
John: "I see."
Mary: "Hello there John. As you can see I am already inside the shopping center; I am behind the guard
Guard: "Yes, she is clearly inside the shopping center."
Mary: "John, allow me to purchase a watch at that store, so that you may enter."
John: "I appreciate it very much."
Mary: "I have returned from the store. Guard, I have purchased a watch; here is my sales receipt."
Guard: "Yes, that is acceptable. John, you may enter."
John: "Thank you for buying that watch for me."
Mary: "No problem. I will enjoy wearing it."
In this case, Mary did not buy John a watch; but surely, Mary did buy a watch for John, that is, for John's benefit. John likely never saw or touched the watch.
However, even with this particular context, this alternative interpretation is strange, and a native speaker would tend to understand your intended meaning. I just introduced this alternative interpretation to illustrate the fact that "bought me" and "bought for me" are not exactly the same.
To summarize, your usage of "for [...]" here is acceptable, though it is slightly more correct to say "bought me {the watch}". It is a slight enough difference that most native speakers would not be aware of it during normal speech.