The situation could dramatically change if President Bryan is strictly neutral and does not give the large amounts of money and arms that Wilson provided to the Entente in OTL. And even if Bryan's foreign policy has no impact on the election (which given how extremely close it was in OTL is something I highly doubt), the fact is that Bryan was seen by many voters as too radical. That's a major reason he lost all three times he ran for President. Wilson by contrast was not seen this way - in fact business leaders supported him over the GOP in 1912 in order to stop Roosevelt. So Bryan is likely to lose in 1916.
Surprisingly, Bryan did not oppose the
sale of arms to belligerents - though (not altogether consistently) he drew the line at allowing loans to facilitate this. See Devlin[1].
Wilson was advised by Lansing that for a government to take power to prohibit private sales of arms would be unneutral, the more so as the power could be used effectively only against one side. Bryan, who might have been expected to take the sentimental view, in fact shared Lansing's opinion.
So the Entente could still have purchased munitions, but would have to sell securities or pay in gold rather than by raising loans. Would such a point really have registered with American voters?
As for him being radical, by 1916 he will have been POTUS for three years, so people
know what his policies are. Are any of them likely to have been much more radical (in unpopular ways) than Wilson's? [2]
Another point is whether his administration will be seen as Southern-dominated, to the degree Wilson's was. As Livermore notes [3], the Sectional issue hurt the Democrats quite a bit in 1916
- - the sectional issue cut down many of Wilson's followers, especially in Illinois, Indiana and Wisconsin, where Democratic loses were heaviest
If Bryan's Administration is less Southern and more Midwestern in character, he and the Democrats are likely to do better in that crucial region. In short, to me it all looks rather "swings and roundabouts".
[1] Patrick J Devlin
Too Proud to Fight, ChVII, p177.
[2] One interesting question is his possible stand on Prohibition. OTL he came out for it only after his 1908 loss, when he saw himself finished as a Presidential hopeful. If he still considers himself one, may he postpone "coming out" on the issue? Or if not, will German-American voters forgive his prohibitionist views for the sake of his antiwar ones? Could be important in a close race.
[3] Seward W Livermore
Woodrow Wilson and the War Congress, Ch1.