Is “puppies” a justifiable translation of κυναρίοις?

Jesus rewards the woman not because she came up with a clever saying, but because her saying, either intentionally or accidentally reminds Jesus of the prophecies that though the Jews come first, the gentiles will come after.

Presuming that Jesus and the woman were more comfortable speaking in Hebrew though it was recorded in Greek:

table comes from a root meaning spread out.

The dog 'KLB' shares the morphed subroot 'LQ' which means to lick, learn, and gather or glean'. And they were used as street cleaners of the things dropped or discarded.

KLB LQQ We have the elements necessary to see the reminder: "The Dogs (KLB) at the door(B) learn(LQQ)".

This is the imagery of the gleaners of the field. Those who are not of Israel and who do not gather in the first harvest, may glean in the field afterward. The small morsels left behind are available to the gentiles and sojourners "under the spreading cover of Israel" (table).

Since the kingdom of Heaven is like leaven (teaching), the reminder that the gentiles are included in the promise is particularly poignant. All she asked for was a small portion of grace, symbolized by the bread (his body) which was broken for us at a time when he (the door) was in hiding (desolation as a symbol of being rejected by Israel).

So placing herself in a position of a submissive gleaner under the cover of Israel, rather than as a ravenous wild dog, puppy may not be precise, but it probably isn't bad. Especially, since in English, we often call even old dogs puppy as a term of endearment rather than an indicator of age.