I suspect that $1{$2} is recognized as addressing an associative array, but I could not find any mention in the documentation about such an expansion in the substitution part of a RE.

See Quote and Quote like Operators:

... In the following table, a {} represents any pair of delimiters you choose.

Customary   Generic         Meaning     Interpolates
              ...
             s{}{}       Substitution      yes*
              ...
* unless the delimiter is ''.
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Where "interpolates" means the same thing as what happens to strings in double quotes, like "$hash{$key}", or, in your case, "$1{$2}". The error message gives you a hint: "Use of uninitialized value within %1 in substitution iterator" - so yes, $1{$2} means the same as $x{$y}, that is, access the hash %1 and fetch the key named by the string stored in $2. Your solution of adding the backslash is fine.

G'day Luc,

Welcome to the Monastery.

The documentation you're probably looking for is "perlop: Quote and Quote-like Operators". Note the "Interpolates" column in the table at the start of that section.

In the next main section, perlop: Regexp Quote-Like Operators", you'll see s/PATTERN/REPLACEMENT/... (you'll need to scroll to the end of that section). That simply refers to "replacement text". I can't see any further mention of interpolating here; although, as indicated via my first link, it does occur.

$ alias perle
alias perle='perl -Mstrict -Mwarnings -Mautodie=:all -MCarp::Always -E
+'
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Just as you can interpolate hash elements in an interpolated string:

$ perle 'my %x = qw{0 a 1 b 2 c}; say "$x{0}$x{1}$x{2}"'
abc
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You can do the same in s/// interpolated replacement text:

$ perle 'my %x = qw{0 a 1 b 2 c}; say "012" =~ s/([012])/$x{$1}/gr'
abc
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The same applies to array elements in an interpolated string:

$ perle 'my @x = qw{a b c}; say "$x[0]$x[1]$x[2]"'
abc
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And, equivalently, in s/// interpolated replacement text:

$ perle 'my @x = qw{a b c}; say "012" =~ s/([012])/$x[$1]/gr'
abc
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This has actually been around for a long time. I use it often and, as it's so ingrained, it took a while to locate the doco.

Here's a very quick-and-dirty example of usage to prepare raw text for HTML:

$ perle 'my %ent = qw{& &amp; < &lt; > &gt;}; say "... < & > ..." =~ s
+/([&><])/$ent{$1}/gr'
... &lt; &amp; &gt; ...
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Further to kcott's post:

$ perle 'my %ent = qw{& &amp; < &lt; > &gt;}; say "... < & > ..." =~ s/([&><])/$ent{$1}/gr'
... &lt; &amp; &gt; ...

luc.bouge:   Also see haukex's article for a very useful technique to dynamically prepare a search (sub-)regex from a "translation" hash like %ent.