Although you can cold smoke in a hot smoker the requirements are actually quite different. Depending on the cold smoke generator you use the limiting factors for cold smoking in something designed for hot smoking / BBQ is how cool you can keep the smoking chamber. With something like an AMNPS or ProQ smoke generator (I prefer the AMNPS) you should be able to keep the internal temperature low enough in your Weber to smoke your cheese and fish. In Backyard BBQ there is also a good idea for cold smoking in a Weber Kettle
https://www.youtube.com/watch?v=TFzzE3VaHmoAlthough I started cold smoking in the Weber I progressed to using a separate unit for my cold smoking. All you really need is a smoke source and something to contain that smoke in. Before I built my purpose built cold smoker I simply used a new metal dustbin with board as a cover. Here is a practical cold smoker solution just using a cardboard box.
http://www.smokingmeatforums.com/t/99316/cardboard-box-cold-smokerWhen trying to choose a single smoker to do what you want there are always compromises that you will need to make.
The Weber is great for hot smoking and is good for cold smoking small items like cheese and fish. Trying to smoke something larger, like a slab of bacon or a country ham and you will struggle.
An offset smoker is great for cold smoking but for a novice can be tricky to get a good even temperature for hot smoking.
A ProQ Frontier or WSM are better all rounders.
If I were in your position I would probably go for the Weber for the hot smoking and also get a dustbin/filing cabinet/cardboard box for doing your cold smoking. Once you understand what you will be cold smoking longer term then you can upgrade to something purpose built.
Some commercial smokers - Like the Bradley - will do both but at a cost. The Bradley biscuits are not inexpensive.