Never breed from an animal with tumors if you can help it. You should cull them from the line as soon as you notice them. They can be passed through the line genetically. There are in fact, lines of mice and rats specifically bred for labs which are nearly guaranteed to grow tumors, so they can research them.
Sad, but even the show breeders will cull important rare genetically gifted animals with tumors, because they can ruin lines.
The puncture technique is smart, if you get dark blood, it's probably a tumor. If you get nothing, or some clear fluid, it's probably a tumor. If you get green, white, tan, or cream coloured bloody gook, it's probably an abscess.