Christina,
I am curious as to how the extension cord from campground power connects to the trailer. With the electrical system already in place there must be a power inlet or some provision for the cord to connect. That would pretty much determine what type of cord to get. I am not familiar with this trailer but it surely does look nice in your avatar!
The Pleasant Valley web site mentions a "7 prong round plug" and I am not sure what that is. The term "landline" is also unfamiliar to me.

Assuming the inlet on the trailer is configured like a 15A plug, two flat blades that are parallel and a round one (may be U shaped) for ground, a heavy duty 15A extension cord should work fine. As others have mentioned a 30A to 15A adapter for the campground outlet would be a nice addition, in case the campground 15A outlet is absent or not working.
As Madjack said the converter should be set up to charge the battery. In my homebuilt electrical arrangement a battery switch connects the battery to the converter. Another possible arrangement is for the battery to always be connected to the converter.
You can check whether the converter is charging the battery by plugging in the trailer AC electrical and powering the converter. Then using a multimeter (sometimes called a volt-ohm meter) check the battery voltage. If it is above 13V, then the converter is charging. Below about 12.5V means the multimeter is reading only battery voltage.
Bill
PS This is a digital multimeter priced at $3.99 that seems a great deal. The cheaper the better.

I have one from Radio Shack that might be about the same price after shipping costs. It will probably take a digital meter rather than analog meter to read the voltage that closely.
http://www.kiesub.com/prostores/servlet/Detail?no=877