Possibly, but it would be complicated
As has been said above, no rule allows you to voluntarily fail a saving throw. However, (as you mentioned) since the Minor Illusion spell has no saving throw, that's not a problem here.
Also, as you mentioned, the only thing that causes an illusion to become faint is "if a creature discerns the illusion for what it is, the illusion becomes faint to the creature." So the only way your illusion could stay solid for you after you cast the spell is if you do not know it is an illusion.
There is one thing working in your favor here: the spell Minor Illusion does not need to be cast in an area you can see (although, like most spells, it cannot be cast through total cover). As such, you could close your eyes, and say something like "Hey, everyone in the party: if you can find a green cloth, please silently put it on top of the blood" then cast the spell. If you open your eyes and see a green cloth over the blood, your DM might rule that you are not sure if that cloth is your illusion, or a real one put down by the party (lying on top of your illusion).
If you were to cast the spell while looking in that direction, and saw the cloth pop into existence, you'd definitely know you were looking at your illusion (since you cast the spell yourself: there is no question where it came from). A DM might also rule that the above trick doesn't work, since the cloth you see will me in the exact arrangement (crumpled where you thought it would be, with the exact color of green you thought it would be) that you imagined while casting the spell. But it is within the realm of possibility.
Personally, I'd recommend just searching with your eyes closed, or cover the view of the blood with your hand close to your face.