I would say the verse can be read on both levels, similar to the following verse:

> Whoever works his land will have plenty of bread,  
but he who follows worthless pursuits lacks sense.  
<sub>Prov. 12:11, ESV</sub>

Somebody who puts effort into his land will have his reward, as the land will yield harvest. Similarly, if you take care of your animal, you will have your reward, since the animal will be healthy, provide food and multiply. This is also explained in verse 12 (the b-part):

> Whoever is wicked covets the spoil of evildoers,  
but the root of the righteous bears fruit.  
<sub>Prov. 12:12, ESV</sub>

It is well possible that this series of proverbs has its origin in agricultural society and may be used in agricultural contexts. However, you may also use it in a more abstract context. Compare the English idiom "we reap what we sow": it can be used in a farmer context, but since most of us aren't farmers it is also frequently used in a more abstract context.

In that way, Prov. 12:10–12 can definitely be read more generally than regarding animal welfare / care of nature alone. Also consider that the verses explicitly refer to the reason for this treatment: you will "have plenty of bread" and "bear fruit". Thus, the reason for this treatment is not some intrinsic value of animals which animal right activists might advocate, but rather a cost-benefit analysis.

As a side note: you note correctly that רדה is traditionally taken as "the word used for harsh rulership". One may wonder whether this meaning does not actually depend on the traditional reading of Gen. 1:26,28. Then applying it there would be circular reasoning. Like [Nigel J](https://hermeneutics.stackexchange.com/questions/33437/proverbs-1210-is-it-really-talking-about-animal-welfare/33439#comment65221_33437), at first sight I'm not convinced the Hebrew implies harshness. In cases in Leviticus (25:43,46,53), פרך "rigour" is added to indicate harshness; in Isa. 14:6 אף "anger". Why would it be necessary to add these words if harshness is implied by רדה? In Jdg. 14:9, the root is used with the meaning "to take", no harshness implied. In the other cases it means to have dominion, but not necessarily harshly so. To make any claims one would have to dig deeper, but at first sight it looks as though the meaning of רדה may have been influenced by a traditional reading of Gen. 1:26,28.