Is it secure to use ECDSA for any arbitrary point on the Elliptic Curve as the Generator point?
However, suppose that you are aware of public key $K$ and are looking to do a tweak of $K$. Let $t$ be the tweaking factor, and so $tK=K'$. If you then wanted to prove that you are in possession of $t$, how would you do that?


The obvious way to do this is a Schnorr proof of knowledge, which does precisely what you're looking for; given a public $K, tK$, it demonstrates that you know $t$.  Wikipedia lists the interactive version; it can be made noninteractive by having the prover generate $c$ based on a hash of the commitment $t$ (and possibly other stuff if you need to bind the proof to something).


  An immediate thought (which I am confident will not work) is to use ECDSA with $K$ as the generator point. However, it might be the case that $K$ is of a very small order $n$ which may not be prime. 


That would also work (which isn't that surprising; ECDSA is essentially based on Schnorr, tweaked enough to avoid patents that have since expired); it's kludgier, however it might be preferable if you just happen to have an ECDSA implementation just lying around.  If $K$ does have small factors, you might need to work around it (as some values might not be invertable), however that wouldn't appear to be a major issue.

And, if $K$ is a very small order, then deriving $t$ would be easy, no matter what proof-of-knowledge method you use (and so there's little point in the problem).  In any case, we almost always use elliptic curves where the order of the curve is $hq$, where $h$ is a small value (the most common values are 1 and 8), and $q$ is a large prime - except for a handful of points (actually, a total of $h$ points), all points have a large order (at least $q$).