PCRE (1043 bytes)
After randomly generated regexes have failed me (the ideas were good, but I couldn't generate adequate problem instances), I've decided to hand craft this one. I dub it "A whole lot of rules to satisfy".
^(?=^([^)(]*\(((?>[^)(]+)|(?1))*\)[^)(]*)*$)(?=^([^][]*\[((?>[^][]+)|(?3))*\][^][]*)*$)(?=^([^}{]*\{((?>[^}{]+)|(?5))*\}[^}{]*)*$)(?!^\(.*)(?!.*\(.{250}\).*)(?=.*\[.{250}\].*)(?=.*\{.{250}\}.*)(?=.*\[.\(\).\{\}.\].*)(?=.*\}...\[...\[...\]...\]...\{.*)(?=.*\(\(..\(\(.{68}\(\(\)\).{43}\)\)\)\).*)(?=.*\{..\{..\{.{65}\{\}\{\}.{33}\{\}.{107}\}\}.\}.*)(?=.*\[\{\{\[\(\{.*)(?=.*\[\[..\[.{6}\[.{6}\]\]...\]\].{6}\[..\]..\[\].*)(?=.*\]\]\}\}\}.\)\)\)\).{96}\]\}\}\]\]\]\}\]\]\].\)\]\].*)(?=.*\]..\).{6}\(.{7}\{.{5}\[...\[.{5}\{\[.*)(?=.*\[.{87}\{.{45}}{.{38}}.{27}\].*)(?=.*\(\{.{32}\(.{20}\{.{47}\].{43}\{\{.{25}\}\}.{18}\].{5}\}....\}.{5}\).*)(?=.*\{.{12}\(.{5}\(...\(...\{\[.\{\[\[.*)(?=.*\{\(.{21}\).{8}\}.{14}\[.{7}\]..\{.{5}\{\}....\}.*)(?=.*\(.\{.{49}\{.{16}\}.{25}\}.{66}\).*)(?!.*\(\{\(\(.*)(?!.*\(\)\[\].*)(?=(.*?\].*?\)){15,}.*)(?=(.*\[.*\(.*\{){5,9}.*)(?=.*\({3}.{105}\[{3}.{105}[^}{].*)(?=.*\(..\).{5}\(\)....\}\}\].\{\{\[.{22}\[.{35}\}\}\].*)(?!.*\(\(.{178}\])(?=(.*\[..\]){8,10}.*)(?!(.*\([^\(\)]{5}\(){4,}.*).{63}(.{6}).{130}\11.{51}$
And expanded:
^
(?=^([^)(]*\(((?>[^)(]+)|(?1))*\)[^)(]*)*$)
(?=^([^][]*\[((?>[^][]+)|(?3))*\][^][]*)*$)
(?=^([^}{]*\{((?>[^}{]+)|(?5))*\}[^}{]*)*$)
(?!^\(.*)
(?!.*\(.{250}\).*)
(?=.*\[.{250}\].*)
(?=.*\{.{250}\}.*)
(?=.*\[.\(\).\{\}.\].*)
(?=.*\}...\[...\[...\]...\]...\{.*)
(?=.*\(\(..\(\(.{68}\(\(\)\).{43}\)\)\)\).*)
(?=.*\{..\{..\{.{65}\{\}\{\}.{33}\{\}.{107}\}\}.\}.*)
(?=.*\[\{\{\[\(\{.*)
(?=.*\[\[..\[.{6}\[.{6}\]\]...\]\].{6}\[..\]..\[\].*)
(?=.*\]\]\}\}\}.\)\)\)\).{96}\]\}\}\]\]\]\}\]\]\].\)\]\].*)
(?=.*\]..\).{6}\(.{7}\{.{5}\[...\[.{5}\{\[.*)
(?=.*\[.{87}\{.{45}}{.{38}}.{27}\].*)
(?=.*\(\{.{32}\(.{20}\{.{47}\].{43}\{\{.{25}\}\}.{18}\].{5}\}....\}.{5}\).*)
(?=.*\{.{12}\(.{5}\(...\(...\{\[.\{\[\[.*)
(?=.*\{\(.{21}\).{8}\}.{14}\[.{7}\]..\{.{5}\{\}....\}.*)
(?=.*\(.\{.{49}\{.{16}\}.{25}\}.{66}\).*)
(?!.*\(\{\(\(.*)
(?!.*\(\)\[\].*)
(?=(.*?\].*?\)){15,}.*)
(?=(.*\[.*\(.*\{){5,9}.*)
(?=.*\({3}.{105}\[{3}.{105}[^}{].*)
(?=.*\(..\).{5}\(\)....\}\}\].\{\{\[.{22}\[.{35}\}\}\].*)
(?!.*\(\(.{178}\])
(?=(.*\[..\]){8,10}.*)
(?!(.*\([^\(\)]{5}\(){4,}.*)
.{63}(.{6}).{130}\11.{51}
$
Tested on Regex101 - depending on your computer you may need to up the max execution time.