- We can see the shrine become big, little by little, from the ferry.
In your example, the difference between the versions that use "become" and "becomes" is that one version uses a non-finite clause complement ("become") and the other version a finite clause complement ("becomes"). Sometimes the meaning of the sentence might be affected. The differences between the two versions can sometimes be seen more clearly if corresponding versions using personal pronouns were looked at. And so . . .
I'm thinking that a practical way of explaining this is by using a diagnostic tool which uses the case of personal pronouns: nominative versus accusative.
First, let's do a little parsing:
- We can see [the shrine become big, little by little, from the ferry].
The expression inside the brackets contains the complements of the head verb "see".
Your problem involves the question: Does the verb "become" have to be in subject/verb agreement with the noun phrase (NP) "the shrine". That is, is the verb "become" a tensed verb form (i.e. a present-tense or past-tense verb form). For if it is, then the verb agrees in number with its subject.
This diagnostic tool involves replacing the NP with an accusative personal pronoun and using it with a non-tensed verb form, and then see if the result still has the intended meaning. In this case:
1.a) We can see [them become big, little by little, from the ferry].
1.b) We can see [it become big, little by little, from the ferry].
1.c) We can see [him become big, little by little, from the ferry].
Those above three versions seem to be fine to my AmE ear. If you agree, then it seems that a non-tensed "become" is acceptable here, and also would be acceptable in your original example: "We can see the shrine become big, little by little, from the ferry." In other words, the head verb "see" in your example accepts a non-finite clause (which is the subordinate clause headed by "become").
Now let's see how the corresponding nominative personal pronouns work with a tensed verb form that agrees with it in number:
2.a) We can see [they become big, little by little, from the ferry].
2.b) We can see [it becomes big, little by little, from the ferry].
2.c) We can see [he becomes big, little by little, from the ferry].
I'm not sure, but these versions sound, er, so-so to my ear--maybe a bit awkward when compared to the #1 versions; though, maybe they could be completely acceptable to other ears. But if we also insert the marker of clausal subordination "that" into the #2 versions:
3.a) We can see [that they become big, little by little, from the ferry].
3.b) We can see [that it becomes big, little by little, from the ferry].
3.c) We can see [that he becomes big, little by little, from the ferry].
then, those #3 versions seem to be better than the #2 versions. Maybe more people will think that the #3 versions are even more acceptable (than the #2 versions). Maybe the #3 versions are just as acceptable as the #1 versions.
For all versions--#1 and #2 and #3--you might want to re-evaluate them with the word "bigger" replacing "big", and see how those versions sound to your ear (don't be surprised at which ones actually sound better). And then, at the end, replace the 3rd person singular personal pronouns with your original NP "the shrine": the evaluations probably should be the same (I would think). Then you could re-run the diagnostic tool with the versions using the verb "becoming" (instead of "become").
CONCLUSION: It seems that the #1 versions, which use the non-tensed verb form "become", are fully acceptable. As for the other versions (#2 and #3), which use the tensed verb form "become(s)", er, well, it's up to you to evaluate them since my ear is now word-weirded out from reading and re-reading these examples and so, can't be trusted.
ASIDE: I've tried to side-step a lot of topics in this post, in order to shorten my answer. Such topics include: Why did I use only accusative personal pronouns when I used the non-tensed verb forms, and why did I only use nominative case with the tensed verb forms? What about testing to see whether the subordinate clause could be a subjunctive clause? Does the verb "see" have one complement or more than one complement? Are versions #2 and #3 called declarative content clauses? Are the #1 versions called catenative constructions? Etc.
CAVEAT: I might have over-generalized some stuff, and maybe made some "typos" or, er, errors--or maybe mis-worded some stuff. Point them out to me and I might try to correct them. If you want any specific stuff explained in more detail, go ahead and ask in a comment.
ADDED: There is an older thread which has some info that is related to your question:
But that thread, and its answer, deals with non-finite subordinate clauses only (which seem to be in a catenative construction). And so, it does not deal with your issue that involves a finite subordinate clause headed by "becomes" (which in your example is a declarative content clause); nor does that other thread provide an explanation of a type that would be appropriate for your students at their level to learn from (imo) -- though that answer is a good answer for that other thread.