I believe your intentions here are good, but I don't recommend trying to rank various types of grief.
Past a certain threshold, all kinds are both the same and incomparable.
Witnessing a cherished part of your life die, while you do not and can do nothing about it, is a uniquely powerless experience. Because, while you continue to live, a part of you dies.
If a person can empathize with this experience in a genuine way, what matter is it their loss?
But I don't think it's a valid criticism here because OP actively calls out they trying to find the most appropriate path for empathy ("support in") while deliberately not equating the two experiences of grief (avoiding "complain in").
If someone did that in front of me I'd probably be compelled to say "Dude, now is not the time for this" or something like that but I can't imagine any person in my group behaving that way.
It’s a tough one, because the grieving mind isn’t particularly rational. You have to be especially careful of things that would otherwise be completely fine.
Much of it comes down to wanting real empathy -regardless of where their understanding of loss comes from. What isn’t desired is any kind of equivocation or comparison. And that is so dependent on how the message is delivered.
And I’ve lost dogs too, that can be absolutely devastating, so intellectually I get it, but the point still stands.