Short Answer: Possible? Yes. Probable? No.
The "advantages" of Synge's translation
First, let's put to rest Synge's claims about the advantages of his translation.
Regarding the consistently personal use of φανερόω, this verb does not have a consistently personal use (e.g. Mark 4:22) -- unless he means in this verse, which would be to commit the logical fallacy known as "begging the question". (If we are trying to determine whether it is personal in both cases, we can't use the "consistently personal nature of the verb in this verse" as evidence in favor of a consistently personal interpretation.)
Regarding the shift in the meaning of ὅτι, the word doesn't have "a meaning"; it is a conjunction with a semantic range and there is no reason whatsoever to favor the same interpretation for each instance of it within a verse. (For example, in 5:9 it is virtually impossible to assign a single "meaning" to the word in both instances.)
Neither of these claims are valid as "advantages" of his interpretation over the ESV's.
Is Synge's translation possible?
The main differences between the two translations are:
He translates it as "he has not yet been made manifest" whereas most take it as "it has not yet been made manifest". The verb itself is not gender-specific, so technically the verb allows for either translation. Syntax and context would be our only way of deciding, but more on that in a moment.
He translates the first "ὅτι" as "because" whereas most take it as "that". Either are possible; again, only syntax and context can help us make that determination.
So grammatically, yes, it is possible. The only way to determine whether it is correct (or probable) is to examine the syntax and context to try to determine which makes more sense with John's flow of thought and writing style.
Is Synge's translation a better translation?
Let's first look at the syntax.
The verb needs a subject (whether supplied or implied.) In Synge's translation he takes it as an implied "He", which is possible, as evidenced by the next appearance of the verb. The ESV takes "what we shall be" as the subject, which is also possible.
The "now" and "not yet" bears consideration. The Greek seems to suggest a contrast between the two (based on the explicit markers and positional priority.) In the ESV "now" we are children, and "not yet" has it been revealed what we will be. The comparison makes sense, as does the following reassurance that though we don't know exactly what we will be, we do know that we will be like Him. In contrast, Synge's translation has "now" we are children, and "not yet" has He been revealed. The contrast doesn't make sense, and so it is dropped. (Even if we try to extend it into the next sentence it's hard to explain: "now" we are children and He has "not yet" been revealed. We know what we will be when He is revealed. Why the explicit and emphatic "now" and "not yet" though? Why not an explicit and emphatic "then" in the second sentence?)
Synge's translation puts "What we shall be" first in a sentence where we would normally expect "We know" to be first. This means he takes it as having positional priority, making it a point of emphasis. But why emphasize "what we shall be" in the second sentence? In his translation "what we shall be" is "like Him". What is he contrasting that with? Being children of God? In other words, his interpretation of the Greek makes it sound like "we will be like Him" versus presently being "children of God".
Just a quick note on context as well: Synge's translation leaves John claiming that "He has not yet been manifested", when he will say just a few verses later that "He was manifested" (3:5).
It seems to me that Synge's translation, though grammatically possible, is syntactically and contextually unlikely. His two (provided) reasons for preferring his translation offer poor support, and his translation seems to create more difficulties and questions than it solves. The only real reason I could see for preferring his translation is to avoid the statement that "it has not yet been made manifest what we will be", which has been a topic of discussion for a long time to be sure... but I'm not sure Synge's proposal is the best answer to our questions about that statement.