The "high power" TDA8560 audio amp you found is obsolete and is not made anymore.
It has exactly the same 4W per channel into 8 ohms at clipping with a 13.2V supply as all the other car radio amp ICs.
Its output voltage swing is also exactly the same as all the other car radio amp ICs. Your tweeters are driven with voltage, not power.
Don't you understand that its "40 Whats output" is only when its speaker is 2 ohms so the current in it is very high (but the current in your tweeters is very low), and when its distortion is a horrible-sounding square-wave so the extra harmonics (but not produced by your tweeters) add to its output power, and when its power supply voltage is very high at 14.4V which makes the IC very hot.
The deer will laugh and say, "What is that little sound?" They might be attracted and go to see, SMACK!