No, Bonus Damage is only delivered when something has a Vulnerability
Vulnerability is detailed in the Basic Rules:
damage of that type is doubled against it
Lycanthropes do not have a vulnerability to Silver, but an Immunity against nonmagical attacks (see below.) The Magical source of damage (or a Silvered weapon) allows that Vulnerability to be bypassed, but they do not deal extra damage.
But what about Electrum?
There currently are no rules in the existing books about the effects of Electrum weapons or Electrum Plating. That falls into Homebrew territory, but given that only Silvered weapons are included in the stat block, one could easily believe that impure Electrum (mixed with Gold,copper, and other metals) does not have a sufficiently pure silver for it to be effective. Had the stat block included Electrum (or other mixed metals) as acceptable, then those would suggest that they might be effective.
Magical sources of damage bypass Lycanthrope Immunity (silver is not required)
It's the +1 aspect of the shortsword that matters here, not the electrum aspect. All Lycanthropes include the following Damage Immunity clause in their stat blocks (emphasis mine) from PHB (pages 208-11) and the updated errata:
bludgeoning, piercing, and slashing damage from nonmagical attacks that aren’t silvered.
Having a silvered weapon also bypasses their immunity, but in the case of your +1 electrum shortsword, it is the +1 that bypasses it without having to worry about the Electrum.
To be clear, the +1 from the Shortsword is not the only damage that bypasses the immunity. The +1 magic weapon makes ALL of the damage bypass the immunity. The weapon is fully magical and delivers full damage as it is a magical source of damage.