One Sledgehammer Script coming up: select your text frame and run this little Javascript. It adds rounded sides as per original frame size and copies line color and thickness from the text frame (where it is removed). Then it groups everything together, and leaves it selected (since you started with a selection as well).

It adds the new objects (rectangle, circles) simply to the parent of the selected text frame -- under normal circumstances, that ought to be the page you are looking at.

textframe = app.selection[0];
left = textframe.geometricBounds[1];
right = textframe.geometricBounds[3];
height = textframe.geometricBounds[2] - textframe.geometricBounds[0];
center = (textframe.geometricBounds[2] + textframe.geometricBounds[0])/2;
radius = height/2;
leftRounded = textframe.parent.ovals.add({geometricBounds:[center-radius,left-radius,center+radius,left+radius]});
riteRounded = textframe.parent.ovals.add({geometricBounds:[center-radius,right-radius,center+radius,right+radius]});
rectBetween = textframe.parent.rectangles.add({geometricBounds:[center-radius,left,center+radius,right]});
roundedRect = rectBetween.addPath([leftRounded,riteRounded]);
roundedRect.properties = {strokeWeight:textframe.strokeWeight,strokeTint:textframe.strokeTint,strokeColor:textframe.strokeColor};
textframe.properties = {strokeWeight:0};
app.select(textframe.parent.groups.add([roundedRect,textframe]));

Wait -- why not make the script do all the work after you resized a frame! Replace the first line with this new fragment (the first line itself is the same, it's the if..stuff that counts):

textframe = app.selection[0];
if (textframe instanceof Group && textframe.pageItems.length == 2)
{
  roundedRect = app.selection[0].polygons[0];
  textframe = app.selection[0].textFrames[0];
  textframe.properties = {strokeWeight:roundedRect.strokeWeight,strokeTint:roundedRect.strokeTint,strokeColor:roundedRect.strokeColor};
  roundedRect.remove();
  app.selection[0].ungroup();
}

It copies the rounded rectangle stroke properties to the textframe and deletes the rounded rectangle, effectively leaving the object ready to be processed as if you selected the original text frame.

Don't use it on a frame that you tinkered with manually, though, as those edits will be lost ... (well, apart from re-sizing).