The word はず is a noun.
- The literal meaning is nock, as in the nock in an arrow. The string of the bow fits well into the nock of an arrow, and this "fitting" sense gradually gained overtones of "naturally this is the way things go", and from there, to "this is the expectation of how things should go". The idiomatic meaning might be something like "expectation".
The word わけ is a noun.
- The literal meaning is split, as in how something splits up or comes apart. This gave rise to a sense of "taking something apart", and from there, "understanding how something comes apart" → "understanding the pieces" → "making a judgment or determination about something". The idiomatic meaning might be something like "way" (as in, "that's the way it is") or "reason", depending on context.
In Japanese grammar, you can't just stick two nouns together and have it work as a relative clause.
When you stick two nouns together, you usually get a new noun -- consider English dog + house, or egg + box, or reason + expectation -- in all these cases, the second noun is the "main" noun, and the first noun tells us a bit about what kind of "main" noun we have. It's the same in Japanese: 犬【いぬ】 + 小屋【こや】 is a kind of koya for an inu, 卵【たまご】 + 箱【はこ】 is a kind of hako for a tamago, and わけ + はず is a ...
Well, I can't quite figure out what a "reason expectation" would be, but it's certainly not what the speaker in the sample text means.
Let's look more closely at the sample sentence and break it down.
- そんな: "that kind". This must usually be followed by a noun.
- わけ: "way, reason". Together with そんな, basically means "like that" or "for that reason".
- ある: "there is". Used here to form the predicate of the relative clause そんなわけある. This whole clause -- not just the noun わけ -- modifies the following noun. As a translation into English, think of this like "that something is/are" -- the "that" is needed in English to indicate the subordinate clause.
- はず: "expectation".
- ない: "there isn't".
From literal to idiomatic English:
そんな || わけ || ある || はず || ない
that kind of || reason / way || there is || expectation || there isn't
there's no expectation that [things are] that way
it just won't be like that / it's just not like that
"couldn't this ... just be 「そんなわけはずない」, with the ある completely removed?"
If we take out the ある, it would turn into something like this:
そんな || わけ はず || ない
that kind of || reason expectation || there isn't
there isn't any such reason expectation
... and I just can't make any further sense out of this.
So no, this doesn't work without the ある.