Frame in photos desappears when save

It is possible to make "export" in JPG format from the Print module, as screenshot below. This method outputs the page layout(s) that you see, including background colour / page border graphics / image border lines / any text labels or watermarking. There are controls on resolution and colourspace but to size the output in pixel terms is a bit indirect.

The page layout is so-many inches wide and high - either natively, or as you override that in the Print Job panel (Custom File Dimensions). Then this-many inches multiplied by a Resolution figure in pixels-per-inch, controls the pixel dimensions of the result. Say you want 1200px x 900px output: you could set the page to e.g. 12" x 9", set a notional Resolution of 100ppi, and that combination will achieve your desired pixel numbers. In the screenshot below, a 1800 x 1500 pixels image will be produced. This would be identical in picture quality to an 1800 x 1500 pixels image produced by setting the page to 18" x 15" and choosing 100ppi Resolution - what will differ is that page elements such as borderline stroke widths etc, are determined by physical measurement units such as inches or points.

The result is one or more JPG files saved into a destination folder, each one displaying one or more photos, depending how you have laid things out. These files will just be numbered sequentially and will not reflect any particular photo's filename nor metadata. Electronic "prints", in effect.

ANOTHER quite different method: you can opt in the Export settings, to have some Postprocessing happen onto each exported photo - adding a border, say. This border will not have been seen inside the Catalog; you would need to define and implement whatever needs to be done. There are utilities that can do this, or a Photoshop action can be encoded as a Droplet and then you can invoke that via the LrC Export settings, to automatically execute some series of actions onto each exported image in turn.