SME 3009 II w/ Linn Sondek

Does the Series II arm mate well with the LP-12? I have friends that have made mention of this combo, but I have rarely seen postings with this setup.
No. The Linn likes arms that are a little bright in character. The classic arm for them was the Grace 707, which I have on mine. It was supplanted years ago by the Ittok and its descendants. The old SME, which tended to sound soft, would aggravate the Linn's deficiencies. The newer SMEs are too heavy for the Linns suspension, a point to be aware of when looking at arms. The Rega arms should match fairly well sonically but I am not sure about the weight. I have used them but before I had a Linn. I have 5 of the older SMEs sitting around but never considered putting one on the Linn. I think if you query Linn they will agree as most of what I have stated was derived from their statements over the years.
I agree with Stanwal. I currently use a Grace 747 on my Dual 1229. When I was using an LP12, I had an Ittok and a Basic Plus on it. It sounded best with the Grace 747. The Grace 707,727,747 family of tonearms are among the best sounding you will find. I wish they were still in production. The SME II is a great arm for an older, rim drive table. It would work great with my Dual, or an old Thorens, Garrard etc.
I had my hands on a 707 recently, but sent it back as it was missing a part. Would a 707 be an upgrade to a Basik
LV-X? If Im not mistaken, I think that the Grace requires a fairly high compliance cart?

I'll avoid the SME - thanks for the info! - but am considering an Ittok, Rega variant or a Hadcock.
It is better then the Basic. In the same mag mentioned later the 707 and Ittok were rated very good and the Basic LVX good +. It is a lower mass than some , to quote Hi Fi Choice no.30 [1983] " Effective mass was low at 6 g. and suitable for medium to high compliance cartridges, and extra weight may be necessary for low mass, low compliance models (otherwise moderate bass lift in the 20-40 Hz region may occur." I would think that low compliance high mass cartridges would work as they increase the mass of the arm. The effective mass is the arm + cartridge + mounting screws. I think the Denon 103 would be one of these. In any case the Denon 304, which is probably the best Denon has made will work, I just bought one for $330 including shipping from Europe. The LVX mass is 12.5 grams so if you want to stay with very low compliance cartridges it will work well. You can add mass to the Grace, many Japanese cartridges use to come with a lead weight to add mass. If I was happy with the arm I might consider one of Linn's endless series of upgrades to the table itself.
Again, Stanwal is right on the money. The Basik is a good arm, the Grace is a sensational arm. Grace manufactured an additonal weight that could be placed on the arms counter weight for low compliance cartridges. One thing about the Grace arm that you will notice immediately, is the SUPER clean tight bass... I always used to think that you needed a large arm (Ittok, SME, Audio Technica, Ortofon, etc.,) to get the huge bottom end, but the Grace proved me wrong. It was wonderful on the LP12, and even better on the rim drive Dual 1229.
Honestly, the Basik, Grace or the Ittok are all fine perfomers. With your table I would avoid the SME, Audio Technicas, Ortofons, and the old Decca arms. Too soft sounding with this table. Now, a Grahm might be a nice match, and for less money the Rega RB-300 is excellent and well matched to the Linn.
My only fear with the Grace 707 is the headshell. I don't know if they are easily replaceable. Perhaps the 747 is better in this department?

One of the Rega variants seems like a good way to go and would most likely perform well on a number of TT's other than just the Linn.

I am also considering a unipivot-type although I have no experience with them.
Roblanger, The Grace headshell is standard. When buying a replacement, you have to make sure that the male end of the headshell that plugs into the tonearm is adjustable. Meaning that the keyway pin needs to be turned toward the bottom instead of the top as in most tonearms. I have found that the better headshells are all adjustable in this regard, usually with a small ALLEN head screw. The 707A has a removable headshell, the 707 does not. 727 and 747 are both removable.