You shouldn't need an anti-virus....
You are right to think that the smart tv is indeed running an operating system and is connected to the internet so therefore in theory it could be infected by a rogue application.
However it is unlikely to need any type of anti-virus and there are a few reasons for this.
1. The type of apps that are in the TV are pre-installed when the TV is purchased, or installed via authorised updates from LG therefore these would have already been checked to ensure no malicious coding is in the apps. For a virus to infect something it doesn't just need a web browser, it needs some form of OS platform to execute it's commands on and an ability to access certain bits to fool the end user - such as creating windows that look like web browsers, or keylogging. The only slight risk is with Java applications running on the TV as many of these are unlikely to be checked properly, however Java apps would struggle to get low enough access level into the TV's own programming to do any type of serious virus damage, at worst it may pop up a screen asking for your bank details, but at this stage alarm bells should normally be ringing as it should look very different to what you'd expect to see on your TV.
2. Most smart tv''s run on some kind of Unix based operating system, possibly even a variant of Linux. Unix based OS's are a lot more resilient to viruses than Windows OS (Windows is getting better with security). In Unix based OS's you have a root user login which really shouldn't be used all the time as these are administrator accounts with full access to everything, and then you have general accounts. If your TV is based on Unix you'll probably find that by default when using the Smart TV section you are logged in to just a normal user account which allows anything running within it all the access you need, whilst blocking out low level access to the TV, in the TV there wont be an option to login to the root account or anything, all this will be hid behind the scenes, and the only way into the root account will be when someone plugs a usb dongle into the USB service port with the right files on it (such as an update file or something), at all other times the TV will just operate in a non-administrator setting which will prevent any malicious software from accessing parts that they shouldn't be able to access. This is why viruses are a lot more rarer and cause less damage on Unix based OS's than Windows OS, also other than through Java a virus would have to be wrote specifically for the internal OS in that TV.
3. Worst case scenario - if a virus was to get into a TV the OS in a TV is so small it can be re-installed in a matter of minutes. You can always do a factory reset which would take a few minutes to re-install the entire Smart TV and remove it all together. Most TV's will probably start from fresh every time you turn on anyway as there is very little a smart TV would need to store anyway, it'll work very similar to a Linux Live CD - this wipes the machine on every reboot and only stores documents and settings and browser history/cookies. Inside Smart TV's they probably have a Kickstart EPROM similar to the way the Amiga's used to boot, and the only thing that will be allowed to re-write that will be the official update files from a USB device (unless you find a hacked one - of course using a hacked firmware then you are opening yourself up to the dangers of malicious code)
Also the chances of you finding an anti-virus for your TV is slim to none as the manufacturer would have to write one specifically for their OS that is in the TV, or one would need to be compiled for that TV.