Some big questions would be, "How much do they want changed to make it look like the other chair?" and "What is the client's budget?" How much are they willing to spend on this?
What you can do would depend upon the answers to those questions.
BASIC ASSUMPTION: What that out of the way, I will assume that they want to keep the price down as much as possible. If that is so, then you probably want to do this as simply as possible, which means: No major frame alterations. (such as re-structuring the wing, and what about the arms?)
To go back to my original assumption, keep the price down, then we would leave the frame of the arms and the wings as they are. So, any modifications would only be in the padding and the cover.
ARM FRONTS: You asked about putting a thick cord around the front of the arm. My suggestion is to get some scrap fabric and do a quick mockup/Prototype of both ways, one way on each arm. Then either decide for yourself which way you like the best or take pictures and email them to the client have them decide (which is usually my preferred method). You can also post the pictures here to get feedback. See my article on prototypes here: http://www.upholsteryresource.com/node/261
THE SEAT: Now, about the seat. You mentioned that the customer may decide to make it a solid seat. If this is the case, yes, as David mentioned, you could rebuild the frame to raise the rails at the bottom of the arms. However, that could be more involved than a low budget entails. So I would suggest just adding foam to make it the correct height. However, that causes some challenges in covering the arms and the back (because the extra height of the seat padding would make it harder to cover the bottom section of the arms and back.)
COVERING SEQUENCE: Normally whenever I recover a chair, I finish covering the deck before covering the arms and the back. However, in the case of this chair with having the higher seat padding, I would suggest, before building up and covering the seat, cover the arms, the wings, and backrest before covering the deck (but only put a couple temporary staples at the bottom covering of each). Then you could cover the arms and the back without the seat padding getting in the way. Following that you would then add the seat padding (probably about 4") and shape it to fit as needed, and put the cover on the seat.
THE LINEN FABRIC: Now, about your fabric. Not all, but many of the very thin fabrics (especially if they are plain colored) show all the unevenness and bumps of the padding. In addition to that, they also tend to leave tiny pull-marks at every staple. Consequently, whenever the client wants a thin fabric I highly recommend putting a lining over the padding first and pulling it the same tightness that you would put the finished covering. This allows you to put the cover on and only pull it tight enough to snug it up over the lining. Because the lining is taking the stress of holding the fabric in place, the bumps of the padding and the pull-marks of the top fabric will be greatly minimized or eliminated.
THE LINING: Now, about the additional cost of the lining. Whenever there is a situation like this, I charge both for the cost of the lining and the additional labor required.
THE CLIENT: Before proceeding with a job I talk to the client about the costs and the overall situation. I generally give her options (i.e. the extra cost of lining and labor OR chose a different fabric, OR.... any other options)