The filing for an LLC has nothing to do with a commercial kitchen. It is simply setting up a legal business entity. Yes you are better off to have an LLC as the name implies it limits your liability shall something happen. In NY even though you set up an LLC that doesn't give you the right to operate a business or sell product. In order to provide product either wholesale or retail you need to have a proper business permit. If you are processing meat and packaging for sale then most likely you will fall under the jurisdicition of the department of agriculture as a meat processor. If you are making an end product for sale or distribution then you fall under the guidance of the department of health. Either way you will be required to prove that all prep/cooking/holding/packaging is done in a permitted commercial kitchen. Dont let this discourage you as there are facilities that rent our commercial kitchen space. Frankly I am surprised the catering company didn't ask if you are permitted. As others have mentioned above they will also hold the liability shall something go wrong. When I first started in order to recieve my catering permit as a BBQ caterer I had to show that we were 100% self contained, practically a commercial kitchen on wheels and that all food was prepped and cooked on site.
I know guys who do catering/cooking without these items and it is a risk they inherit. Have something go wrong or get caught by the dept of health and you can all but kiss any future in the food biz goodbye. But there is a cost associated by doing everything legit, and only you can determine if the work from the caterer would justify spending the money..