Really, there was no reason for La Follette to run a hopeless third party campaign for president in 1916--he wanted to get re-elected senator that year as a Republican, at least a nominal one. I added those last five words because he made it pretty clear that he favored Wilson. As David Sarasohn remarks in
The Party of Reform: Democrats in the Progressive Era (University Press of Mississippi 1989), p. 216, La Follette in 1916
"came as close to violating his policy of grudging silence about Democratic progressivism as he ever would. He publicly backed neither candidate but praised the Underwood Tariff and the Adamson [eight-hour day for railway labor] Act in his own reelection campaign. [Hughes of course had vehemently denounced both laws.] Democrats carried out their role in the unique relationship solicitously. As before, large numbers of them crossed over to vote for him in the Wisconsin Republican primary. Aferward, the Democratic candidate running against La Follette urged Wilson not to endorse him against the incumbent, and Bryan, campaigning in Wisconsin, made no mention of the Senate race. Although La Follette remained silent, the Madison
State Journal, edited by a close ally, endorsed Wilson in late October. 'Enemies have unkindly remarked that La Follette was sitting on the fence,' noted the
Nation. But the fence would seem to be located fifty yards inside Woodrow Wilson's lot.'" [1]
Likewise, John Milton Cooper writes in
Woodrow Wilson: A Biography (p.355) that "Among Republican insurgents, only La Follette tacitly supported Wilson, as he had done in 1912; the rest backed Hughes, though often without enthusiasm."
http://books.google.com/books?id=lxoOdaCDbpEC&pg=PA355
That La Follette's courting of Wilson voters was successful is shown by his re-election, where he won 59.23 percent against 31.90 for the Democrat and 6.85 percent for the Socialist.
https://www.ourcampaigns.com/RaceDetail.html?RaceID=46908 Meanwhile Hughes won the state by a much narrower 49.39 to 42.8.
https://en.wikipedia.org/wiki/1916_United_States_presidential_election_in_Wisconsin (That Hughes did nevertheless win WI may show that some German-Americans were still angry with Wilson despite the "he kept us out of war" campaign and the belligerent TR's support for Hughes. Or maybe they were just Republicans who voted for Taft in 1908, split their votes between Taft and TR in 1912, and returned to the GOP fold in 1916.)
I do not know whether Wisconsin law in 1916 allowed one to run for both the presidency and the Senate. But even if it did, running as an Independent for Preisdent and a Republican for the Senate would be a little awkward. (It might for one thing induce the "Stalwarts' or conservative Republicans to run a candidate against him for Senate in the general election.)
[1] "The Republican candidate for United States Senator from Wisconsin does recall, when reminded of the fact, that a man named Charles E. Hughes is running for the Presidency. When Mr. Hughes was in Wisconsin three weeks ago, La Follette was prevented from meeting him by a previous engagement. The Republican candidate for Senator was probably too busy boning up his speeches defending his vote for the Eight Hour law, the Underwood Tariff bill, and the Seaman's bill. Enemies have unkindly remarked that La Follette is sitting on the fence. But the fence would seem to be situated fifty yards inside Woodrow Wilson's lot. Any other year the case of La Follette would have been cited as an example of how hard it is to be a Republican. This year, on the other hand, it only proves how easy it is to be a Republican. One may approve of the Eight-Hour law, of the Underwood tariff, of the Child Labor law, of the Federal Reserve law, of Wilson's foreign policies, of all Wilson's policies, of Woodrow Wilson himself. Nay, one may even vote for Woodrow Wilson and be a Republican. No other party in history ever offered such liberal inducements to its adherents..."
https://books.google.com/books?id=6ec4AQAAMAAJ&pg=PA884