What tuning is *requiring* you to use a 6+ fret? Playing Chord-Melody or Melody Stringstyle? The more or less correct answer is to change tuning tothe onethat does not require the 6+ fret. If it's DAd that requires the 6+, then tune to DAA and play the tune; or vice versa.
I would not suggest a toothpick as a temporary fret - it's too soft, and the note will not sound clearly. Use a length of large paperclip wire instead.
The simplest work around is to just not play the 6+ note, but either rest for that time or play the previous note again or the following note twice.
Since your dulcimer does not have a 6+ fret I would seriously suggest learning to play in DAA rather than DAd. If you must play Chord-Melody style, you can play the same chords, more or less, in DAA (but not at the same frets). But a traditional (no 6+ fret) diantonic dulcimer is probably best played traditionally -- Melody & Drone or Noter & Drone style.