You can do this with awk:
awk -v RS=" " '{sum[$0]+=$0; if($0>max) max=$0} END{print sum[max]}' <<<"${array[@]}"
Setting RS (record separator) to space allows you to read your array entries as separate records.
sum[$0]+=$0; means sum is a map of cumulative sums for each input value; if($0>max) max=$0 calculates the max number seen so far; END{print sum[max]} prints the sum for the larges number seen at the end.
<<<"${array[@]}" is a here-document that allows you to feed a string (in this case all elements of the array) as stdin into awk.
This way there is no piping or looping involved - a single command does all the work.