How to win the machete proving grounds?

You should head straight to the middle to start off with, and take out as many of them as you can. After that you should go the squirt spawners, and finally move on to the towers. I'll also point out that one charged machete throw will destroy a tower in one hit, which makes it preferable over melee since they cower after being attacked. Any coward you missed at the start that you come across while taking care of the spawners and towers should be destroyed on the spot, and the rest should be hunted down after you take care of the towers.

For upgrades, you should use the tier 1 serrated edge upgrade, it's effects not only makes it harder for the squirts to get away, but could still result in them dying if they do. It's effects are:

Attacks cripple foes
Attacks deal damage overtime

For tonics, you should equip Werewhisky, and suicide off the edge of the map until it activates. Make sure you do this before picking up the Machete:

+100% Critical Hit Chance
Note: Active Only When Below 33% Health

You may find this video helpful as well if you prefer visual aids, which uses the same steps outlined by DMA57361:

Get to the middle - where the coward squirts spawn - as quickly as possible and hack away like mad.

The objective here is to kill as many coward squirts as possible as they are spawning, because they are extremely time consuming to pick off one by one if they spread out (as you have already found).

Once they stop spawning, you should next concentrate on the spawning points for the other squirts, then finish up the towers and remaining squirts.

You should be able to complete this challenge without having to upgrade the weapon.