I live on an island where most of the roads are not paved(clay rich soil) and all roads are limited to 25/mph. Also, most roads are mountainous.
I have a 78' W123 manual 4 speed "240D" swapped with a turbo diesel with a 3.07 diff.(thinking of putting the 3.69 back in given the speed limit, I want the lower gears). The W123 has factory springs sold to the African market. The springs give a ride height to the vehicle of about 2-3". It has new bfgoodrich 205/75/15 off road tires and factory oil pan skid plate.
The Question is: Should I weld the Diff for better traction given the situation? I have a fully equipped shop at my disposal.

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In those conditions, that's a "fair weather" car. Welding the diff will make it incapable of going around corners on paved roads and will put a lot of strain on the half shafts. If you have traction problems, you're likely better finding something with AWD or 4WD.

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Current stable:
1995 E320 157K (Nancy)
1983 500SL 125K (SLoL)

Gone but not forgotten:
1986 300SDL (RIP)
1991 350SD
1991 560SEL
1990 560SEL
1986 500SEL Euro (Rusted to nothing at 47K!)

Gone and wanting to forget:
1985 524TD 167K (TotalDumpster™) [Definitely NOT a Benz]

Have a look at this:
http://www.peachparts.com/shopforum/performance-paddock/389941-intrest-185mm-lsd-diffs.html

Does the State of California know that Temecula is an island?

I no longer live in Temecula, I live on an island "26 miles across the sea"

I did that on my 84 280e euro dedicated autocrosser. Turning was interesting with the inside rear tire hopping about at low speeds. Turning at low speeds on the autocross course induced push especially after the Hoosier slicks had some wear on them. I'd think with no paved roads you'll be fine.

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[SIGPIC] Diesel loving autocrossing grandpa Architect. 08 Dodge 3/4 ton with Cummins & six speed; I have had about 35 benzes. I have a 39 Studebaker Coupe Express pickup in which I have had installed a 617 turbo and a five speed manual.[SIGPIC]

..I also have a 427 Cobra replica with an aluminum chassis.