As a researcher, this type of feeding generally wouldn’t be allowed in a controlled scientific setting. There is demonstrable stress and suffering occurring for the prey animal (reptiles and amphibians can certainly feel pain).
Not to mention many animals can and will fight back if you try to feed them to another animal. This is why, for example, it’s considered unethical to feed live mice and rats to snakes: in the wild the snake can often ambush the rodent at its leisure and minimize injury to itself (but notably it is still at risk), whereas in a captive setting the rodent can kick and bite the snake and cause serious injuri0es.
I think sometimes we can tend to devalue invasive species a bit in terms of their suffering. So a good question is: would you feel that the level of suffering was/wasn’t justified if feeding a native species to a heron (a native frog for instance)? If that wouldn’t feel ok, I wouldn’t feed them.
This is actually something I think might need to be discussed more often. There’s been this discourse that invasive species are worthless, but then that gets extended to all introduced species, whether or not they are actually harmful or whether or not they dispersed to the environment naturally or by anthropogenic means. There’s been a general trend in biology that all species that weren’t there when scientists first started surveying the area are pests that must be exterminated, and anything you do to them is acceptable, no matter how cruel and inhumane. I remember in mammalogy classes someone claimed that coyotes and red foxes were non-native and invasive in the United States, and it provoked a huge argument specifically because of this logic that “if they’re invasive, they aren’t legitimate animals and don’t have a right to exist”. I’ve even seen biologists and field researchers kill invasive organisms in horrifically cruel ways. This doesn’t even get into how someone might inhumanely kill a native species if they mistake them for an invasive. Reminds me of the creepy eugenics-esque “cleansing” that was done to try and save the red wolf, where in the mid 20th century they killed ~75% of the surviving red wolf population (mostly in zoos) believing them to be “tainted” by coyote and gray wolf DNA, only for it to turn out that red wolves originally evolved in a hybridization event between Canis latrans and C. lupus. Or how people argue whether the red wolf is a “legitimate” species that should be allowed to exist today.
What gets really creepy is how this dehumanization (de-something, I don’t know what the word is when it applies to animals) can easily get extended to humans, given how our species was an African-Middle Eastern endemic until about 90,000 ka and our species migrates so frequently. From a certain perspective it’s kind of hypocritical for our own species to consider the existence of any introduced species illegitimate. I’ve actually seen some biology professors with PhDs refer to specific ethnic groups of humans as “invasive species”, which is really creepy and I suspect comes from the fact that they were specialists in invasion biology (and thus this sentiment is pretty much an extrapolation of the mindset mentioned above). There are a lot of invasive species that are destructive and must be controlled for the sake of ecological health (garlic mustard, sea lamprey, alewives, house sparrows), but the current general attitude encouraging a lack of empathy for other organisms is concerning.
Euthanizing animals before feeding them is certainly an option (but again, with realizing that incidental taking of a few invasive individuals isn’t going to have a serious positive impact). However, the rules on euthanizing wild animals vary very widely from place to place and you could open yourself to the risk of getting in trouble for animal cruelty depending. As @fishkeeper notes, blunt force trauma to the head is a very quick and relatively painless way of killing vertebrates. In my opinion, it would be ethically preferable to feeding a living, pain-feeling organism to another if the sole goal is death of the focal individual. I’ve done this myself with injured animals (lizards/snakes hit by lawnmowers or cars but still alive), but it’s not pleasant (even if I think I’m doing what’s best for the individual animal).
Not to mention the way most animals kill their prey tends to be excessively cruel. Not because they want to be, mind you, but rather because natural selection doesn’t build animals for maximum efficiency in combat and killing, only what works well enough for the animal to be fed (and to be fair even the most humane ways that humans have invented for killing such as guillotines or lethal chemicals have been found to still cause a great deal of suffering). A lot of animals kill their prey by eating it alive, dismembering it limb from limb while still alive, or both (as in the case of crocodilians and most canids). About the only animal that regularly kills its prey with a minimum of suffering are felids. Cats are known to torture what they catch, but their hunting strategy lends them to kill prey with a minimum of suffering when they decide to get serious. I had a box turtle that I had to feed a fish to in order to give him antibiotics, and he killed it by dragging it out of the water, ripping its eyes out, and then waiting until it suffocated to death before eating it (needless to say, I vowed to never feed him any vertebrates again out of ethical concerns). Nature tends to be rather messy.
And yes, exposure to benzocaine and other oral anesthetics followed by freezing is a fairly humane way of killing amphibians. You should not dispose of any resulting bodies anywhere outside however, as that risks poisoning any animals which feed on the dead bodies.
As an addendum: never, ever try to euthanize vertebrates by freezing them. I’ve talked to some vets and they’ve said they’ve had cases where ignorant pet-owners have tried to euthanize iguanas and turtles by putting them in a freezer overnight, only to find them still alive when they come back the next day.