Japt, 19 14 13 10 9 8 bytes

Currently the best-scoring solution that fits in an answer!

"(()()...()()<"q>)m(l)m(d)q

where the string at the beginning is:

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The total byte count is "only" 6694 3394 3354, so you can try it online!

Explanation

The actual method used is explained in other answers: Split on <>, map each run of parens to chr(len(x)), join again on the empty string. Here, it's the golfing that's the fun part.

Before "golfing", the original code might look like

"..."q"<>" mZ{Zl d} q""

which is a quite literal description: "...".split("<>").map(Z => Z.length.toChar()).join("") Now we need to minimize non-bracket chars. How? Well, first we can do some actual golfing:

"..."q"<>" ml md q

This roughly represents "...".split("<>").map(Z => Z.length).map(Z => Z.toChar()).join().

Now we can abuse the confusing way Japt treats parentheses. ( represents going up one level, as in most languages, but ) represents going down two levels (a space goes down one level), which means we can optimize the code to:

"..."q("<>")m(l)m(d)q

This code acts just the same as the above, but uses two fewer non-bracket chars.

Additionally, if an operator is the first input to a function, it gets turned into a string so that the function can decide what to do with it. This means that we can avoid the quotes if we just make every run of parentheses 1 byte shorter, and split on > instead (with some clever arrangement to cancel out the resulting >) in the code):

"(...<"q>)m(l)m(d)q

This saves us a further two bytes, since we took out two of the quotation marks.