Ruby, 99 89 102 bytes

->y,m{(1..4).find{|d|(A=Date.new y,m,-d).wday<5}
A.strftime"%A %d#{A.day>30?'st':'th'}"}
require'time'

Try it online!

Explanation

Anonymous function tacking year and month as numbers.

We find which of the last 4 days of month ( checking backwards using -d ) are wdays (week days 0-indexed) less than 5 saving each time date object in constant A for later use.

When it's found loop is stopped so A is the last weekday of the month.

Then we format the output, we can only have 31th or xxst

Ruby, 99 89 102 bytes

->y,m{(1..4).find{|d|(A=Date.new y,m,-d).wday<5}
A.strftime"%A %d#{A.day>30?'st':'th'}"}
require'time'

Try it online!

Explanation

Anonymous function tacking year and month as numbers.

We find which of the last 4 days of month ( checking backwards using -d ) are wdays (week days 0-indexed) less than 5 saving each time date object in constant A for later use.

When it's found loop is stopped so A is the last weekday of the month.

Then we format the output, we can only have 31st or XXth.

Ruby, 99 89 117 bytes

->y,m{(a=(Date.new(y)..Date.new(y,m,-1)).reject{|d|d.wday>4}[-1]).strftime"%A %d#{a.day>30?'st':'th'}"}
require'time'

Try it online!

Fixed restricted output at huge cost.

Never thought Ruby had ranges also for dates!

Explanation

Anonymous function tacking year and month as numbers.

We pass the [First day of year .. last day of month(-1)] range and reject wdays(week days 0-indexed) greater then 4.

We return the last element.

Ruby, 99 89 102 bytes

->y,m{(1..4).find{|d|(A=Date.new y,m,-d).wday<5}
A.strftime"%A %d#{A.day>30?'st':'th'}"}
require'time'

Try it online!

Explanation

Anonymous function tacking year and month as numbers.

We find which of the last 4 days of month  ( checking backwards using -d ) are wdays (week days 0-indexed) less than 5 saving each time date object in constant A for later use.

When it's found loop is stopped so A is the last weekday of the month.

Then we format the output, we can only have 31th or xxst

Ruby, 99 89 131 bytes

->y,m{(a=(Date.new(y)..Date.new(y,m,-1)).reject{|d|d.wday>4}[-1]).strftime"%A %-d#{%w(th st nd rd)[a.day%10]||'th'}"}
require'time'

Try it online!

Fixed restricted output at huge cost.

Never thought Ruby had ranges also for dates!

Explanation

Anonymous function tacking year and month as numbers.

We pass the [First day of year .. last day of month(-1)] range and reject wdays(week days 0-indexed) greater then 4.

We return the last element.

Ruby, 99 89 117 bytes

->y,m{(a=(Date.new(y)..Date.new(y,m,-1)).reject{|d|d.wday>4}[-1]).strftime"%A %d#{a.day>30?'st':'th'}"}
require'time'

Try it online!

Fixed restricted output at huge cost.

Never thought Ruby had ranges also for dates!

Explanation

Anonymous function tacking year and month as numbers.

We pass the [First day of year .. last day of month(-1)] range and reject wdays(week days 0-indexed) greater then 4.

We return the last element.