How can I reverse engineer a commercial cleaning solution's "other ingredients"?
Reverse engineering, or deformulation, is a common industry practice. This is a very common type of request for commercial analytical laboratories, and is generally not cheap. You can discover just how widespread it is, and how many commercial providers there are with a quick Google search.

To get yourself a reasonable starting point, you should request a product MSDS from your suppplier. All suppliers should be able to provide a full MSDS, which will give an approximate ingredient listing and component concentration. Sometimes, you will find ingredients listed as "Trade secret" or "Proprietary", but many times you will get more information. It is very likley that your lens cleaner solution is, in fact, predominantly distilled water; perhaps up to 90-95%. However it is the other minor ingredients that are the active ingredients, and may include alcohols or glycols, anti-fog and anti-static agents, anti-baceterials and preservatives. Examples of lens cleaner MSDS can be found via a google search, and this may give you a useful bit of background information.

Note:
1. all suppliers are legally bound to providing MSDSs upon request in the local idiom (official language).
2. this document is safety-based which means it will (a) provide (useful) information on possibly hazardous materials, (b) assess the hazardous component(s), their type, level and correlate to concentration.
3. websites such as state Chemical authorities (ECHA, FDA, ...) publish ratings, studies and lists (of registered/authorized/forbidden) of chemicals for use in particular applications such as food contact, medical, pharmaceutical devices/products.

The next step is to determine what information you really are after, as this will determine the budget you will need to outlay. If you want to just determine whether the solution is predominantly water, you could try a few things yourself (if you are experiment-savvy). Try smelling it, as a first step; check for evaporation if you run a thin film on a glass slide, test density or melting point. These will all tell you much the same thing, which is, yes, the solution is predominantly water. A simple commercial 1H NMR analysis will show you this also, and confirm water concentration up to about 99.9%. This means minor ingredients will show up, but as long as they make up about 0.1% of the ingredient composition, and they contain hydrogen (sodium bicarbonate will not show up for instance). This method is commonly used when you have a reasonable idea about the identity of the minor ingredients, but can involve extensive analysis if you really are shooting in the dark. 

Beyond this, if you actually want to determine the ingredients, and have no idea what they might be, you will need to head down the avenue of GC-MS and/or LC-MS; a combination of chromatography to separate all of the ingredients, and then Mass Spectrometery to try to determine the mass of each component. This type of analysis is generally charged by the hour, and the more variables you are trying to evaluate, the more expensive the procedure. Generally, when analysing a competitor's products, a reasonable knowledge of the types of raw materials exists, and fingerprinting chromatography and NMR traces against raw material libraries saves a lot of time (and money). Sometimes, other techniques such as FTIR, AAS, UV-Vis or ICP may also be used, but these would really only be if the MS methods failed to give you suitable leads.

Commercial analytical labs will be able to give you a very good summary of the best techniques for your sample, but be prepared; the very first question you are likely to be asked is "what is your budget?". Good luck.