Before we deal with your important dilemma, let us point out a few things. First, you should confirm that you are correct in your assumption that you are clearly exempt by presenting the details to an expert on Choshen Mishpat (monetary Halacha). These halachot can be complex, and the way you view the case may be subjective. Second, from a very different perspective, consider that although your acquaintance may not presently be aware of who caused the damage, the truth may come out, and the troubles you are trying to avoid may be compounded significantly later on.
We will now deal with your question as asked. Since we have not found an explicit source on the matter, we will compare your case to related concepts.
The Shulchan Aruch[1] says that if Reuven is financially obligated to Shimon, Reuven must not avoid payment with the hopes of pressuring Shimon to agree to a compromise. However, the Tumim[2] says that if Shimon owes Reuven a corresponding amount but Reuven cannot prove it, he may be evasive in order to achieve the correct result. Avoiding the need for an argument and a possible din Torah when one knows he does not owe money is no worse than the Tumim’s case, and we can thus assume that he would certainly not require you to present yourself to the other side. The Pitchei Teshuva[3] cites a distinction between active obstruction, which is forbidden, and using convenient non-action to gain advantage, which could be permitted. Here too, you are discussing a non-action – not coming forward to admit involvement in the damage – which should certainly be permitted according to the Pitchei Teshuva.
The gemara[4] prohibits making fallacious claims in beit din as a means to eventually win a case, even if one’s vindication is deserved. Such claims would be a violation of the command “Distance yourself from a word of falsehood.”[5] [6] If there is no lie of any type, however, it appears that if one can attain what he deserves simply by not coming forward, he may employ that tactic.
There is an important rule (although one that is difficult and dangerous to apply) that under certain circumstances, one may take the law into his own hands rather than go to the trouble of involving a beit din.[7] Based on the guidelines found in that context as well,[8] there should not be a problem in simply failing to volunteer information and thereby avoiding having to contend with the associated hardships in assuring your rights.
One set of sources that seems to contradict the above indications about not having to volunteer information has to do with the severe steps that are taken against one who refuses to submit to adjudication when so requested.[9] How can we punish someone if he claims he owes nothing and, for all we know, is correct? The answer is simple. The defendant is censored not for his refusal to fulfill an obligation toward the plaintiff and present himself for adjudication, but rather for his offensive behavior to society, as well as his affront to the Torah. Since a Torah society (or any society for that matter) must have an effective court system, once the court determines that one is required to submit to adjudication, he must not be allowed to snub the system and expose it as toothless. Indeed, only if beit din made its credentials clear can a defendant be sanctioned for not submitting, as the sanctions are meant only for those who reject the calls of those bodies that they know have legitimate authority.[10]
Thus, dependent on the caveats presented in the beginning of our presentation, you are not required to volunteer that you damaged your acquaintance’s property if there is a concern that unjustified difficulties would ensue.