I rather routinely toss off cuts - I don't think I've ever had an infection from one (and a bunch of us know how old I am, this is a serious lack of skin infections over a lifetime). But, there is the possibility of tetanus (Clostridium tetani?) and other soil clostridia, not to mention the uggy googys out there now, like MRSA.
Me, I'd check with a doc, if you have one to just go in and ask. Worth the caution.
I've also cut a finger with a kitchen sharpie while cutting a pork roast. I remember staring at it and hanging the finger under the faucet and sort of quietly crying out to my husband about taking me to the doctor. Yah, I'd cut into the tendon, which involved this emergency room dip in, yes, betadine, and stitches. Decades after, the finger a...l...m...o...s...t straightens out as well as it's opposite number on the other hand.
I figure you are doing similar, but hell, I'd still check it, as it IS infected.
Oh, on tetanus. Don't know about current t. shots. In my day, I think they had a seven year span, but it was understood that was really about fourteen. But that was a long time ago. I think you're fine on it, but... don't trust me.
Thing is, I also remember all those clostridium-of many-kind stories.
I also remember bacterial endocarditis stories.. but not sure re which bacti besides strep. (It's been a long time.)