体を張ってでも反対する means that the one who 体を張る is identical to the one who 反対する while 体を張っても反対する means that those two are different.
体を張ってでも反対する： I would rather risk my own body to oppose it.
体を張っても反対する： I oppose it even if you risk your own body.
couldn't 体を張っても反対する be interpreted as "even if I risk my life (body)"?
Practically yes, but that sounds ...