In the case you describe, I would apply my reversibility test: If I do it now, can I undo it? If I don’t do it now, can I do it later?
A lot of things become simple and clear when I apply this test.
You can certainly postpone this decision until a time when it won’t cause you such a pang. Unless you desperately need the space it occupies or have to evacuate and travel light, just let it be until the time comes when you’re okay with letting it go. You’ll know when that happens. Meanwhile, you do not owe it to some other dog.
You could even wrap it up in something and label it, so it’s out of sight for a while. Then, later on, either you’ll be fine with tossing it without ever seeing it again, or you’ll open it up and say, “Oh, I’m so glad I kept this! It still smells a little bit like Spot.”
Despite my natural tendency to hang onto things too long, I have sometimes given something away prematurely or impulsively, especially when I’ve been in some sort of transition. I’m nearly always sorry I did. In particular, I often think of the mahogany dining table that I gave away in haste when I moved west. It had belonged to my parents, and it was big and heavy, and I never thought I’d miss it, but I do. I gave it to someone I hardly knew, so I never even got to hear that she enjoyed having it.
If I’d kept it, my family would still be using it today, and I would enjoy the sense of tradition around it.