Your meter is probably reasonably close to correct. :)
You really don’t need a big amp unless your speakers are hard to drive, and you don’t need watts so much as current.
Current sometimes, and often not, comes with bigger amps.
You might want to look at an online power / dB calculator. For 10dB louder you need 10x more power, so to go from 87 to 97 you would need 10 watts, from 97 to 107 you’d need 100.
Of course it is important to note that those meters are slow responding. Instantaneous transients will peak at higher outputs.