Unless you suspect an improper business practice or conflict of interest (see FAR Part 3 for more information), this is probably more of a training issue than anything else. I could not find in my research of the FAR and DFARS, any requirement for concealing government budget information, but there is some guidance on independent government estimates. It was unclear from reading your question if the purchase was for construction, supplies or services. If the purchase is for construction, FAR 36.203 states the independent government estimate (IGE) shall not be disclosed except as permitted by agency regulations. And, if it's for services, the DoD Independent Government Cost Estimate (IGCE) Handbook for Services Acquisition states that "It is imperative to remember that the IGE is a procurement-sensitive document and should be handled as For Official Use Only (FOUO)". And, generally speaking IGEs and IGCEs are normally handled as FOUO. Although IGEs and budgetary information are not the same thing, it's helpful to know what the FAR does specifically put restrictions on disclosing; things like IGEs and the contractor's proprietary proposal information.
While the FAR doesn't specifically address controls on the handling of budgetary information, I along with probably many others in the government would be reluctant to share that information particularly when in negotiations with the contractor. So, I was a little surprised when I found out that some budgetary information is already available in the public domain. For example, when Congress passes the MILCON appropriation each year, it lists line-by-line the total dollar amount budgeted for each construction project being funded. The Air Force, Navy and Army provide this information to Congress via DD Form 1391s. A DD Form 1391 must be completed for each project submitted, and the form includes a detailed description and cost estimate for the project. Then, Congress can use the DD Form 1391s to rack and stack the projects and decide which ones make the cut to be included in the appropriation. So, a construction company interested in bidding on a particular project can find out what's budgeted for it by looking in the appropriation law that's passed. Also, the topline budgetary information for major defense programs is provided in the NDAA (e.g. 9.4B was in the FY22 budget for JSF procurement).
With that said, I can understand your concern with the contractor getting their hands on the installation's budget information particularly if the contractor you're negotiating with is sole source. I also get wanting to keep this information confidential for negotiation purposes if possible (assuming the contractor would not otherwise be able to get this information if were not for the actions of the individual who disclosed it to them). However, each negotiation is different from the last one, and each government negotiation team may decide how to handle this information differently than another team. It depends on a lot of things (e.g. our relationship with the contractor, the experience we've had with negotiating with the contractor and the quality of their cost/pricing information, etc). Therefore, it depends on the circumstances and what information the team decides may/should/must be disclosed in order to reach a negotiated settlement. Also, your contracting team, with the assistance of your legal advisor, needs to weigh the seriousness of this information being disclosed to the contractor (in other words, how much damage would it really cause). Also, try to assess what was the intent of the person that disclosed it. If it doesn't look like it was done for personal gain or unethical purposes (See FAR Part 3), then it's probably just a mistake. Perhaps, it's due to their inexperience, overzealousness or simply misunderstanding local policy/agency regulations (or the sensitive nature of this information). But, these type of problems can normally be handled through training or retraining.
I assume this individual is a contracting officer representative (COR). If you decide that training or retraining is appropriate, I strongly recommend reviewing with the individual what their duties and responsibilities include, the limits of their authority, when the contracting officer needs to be contacted/involved and what actions must be performed by the contracting officer and ONLY the contracting officer. Furthermore, the appointment letter from the contracting officer needs to spell out exactly what authorities are being delegated to the COR (and which ones are not) in accordance with FAR 1.604. I would also remind the person(s) involved that all exchanges after solicitation release must go through the Contracting Officer in accordance with FAR 15.207(f). Contract negotiations is definitely one of those types of exchanges that must involve the contractor officer. In fact, normally the contracting officer is the lead negotiator and is always, always, always a participant during negotiations. Finally, I strongly recommend discussing your concerns with your legal advisor and possibly even your ethics official.