You do not need a UA code to book a flight on United. United (or any other airline) can sell you a ticket using other airline's codes, provided that they have an interline ticketing agreement with them. Whether or not UA will choose to do this (either online or via the phone) appears to be route- and agent-specific.

However, not all fares can be booked on UA. TK can (and does) limit the fares that can be used by other airlines to book its flights. Taking a look at the fare rules for a sample E-class itinerary, I see: UA will not be able to sell you this fare; it is likely that any fare they can sell will be significantly more expensive than E class.

Having said that, if you're planning to upgrade the flight using TK miles, you may find that it needs to be TK ticket stock anyway -- make sure you check their rules before purchasing your ticket. Also, if you have UA miles, it's only 160K for a roundtrip business class ticket from LAX to MLE; plus, you can get a free stopover along the way. Unless you're sitting on a ton of TK miles with little other use for them, 130K seems pretty steep for an upgrade.

And, finally, as sbm12 mentioned, the 7x is a multiple of the PQD. I see a $1390 fare for E class on ITA; if you were able to book that on UA, the PQD would be $1291 (base fare plus fuel surcharge; all taxes and fees are excluded). So, as a silver, you'd earn 7 x 1291 = 9,037 miles round-trip. If you buy the ticket from TK, you'll earn redeemable miles via the chart on the page you listed, which shows 75% earning for E (not 100%). That comes out to about 15,900 miles for your itinerary. (You do get 100% premier-qualifying miles toward achieving premier status).

If you want 100% earning, you'll need an O-class fare or higher. On the dates I checked, the best option was an S fare, at $1480. So, you'd pay $90 extra in order to get an extra 5,300 redeemable miles; up to you whether or not that's a good value.

Either way, the TK ticket is the clear winner here.