While the answer is yes, it is nearly impossible and depends on a lot of DM intervention and interpretation.
Your Barbarian can change the type of damage they do with their melee weapons.
Method 1: Appeal to the power of Wish
A wish spell can be used to temporarily transform your weapon into an energy type weapon for about 8 hours, which is in line with the same power scope as "grant up to 10 creatures you can see immunity to a single spell or other magical effect for 8 hours" and producing a non magical item worth up to 25,000 gp permanently.
This would thus create a temporary magic item, probably under 25,000 gp in value, of the rare/very rare class. Since the object doesn't last more than 8 hours before reverting, it isn't truly creating a magic item, just a temporary magic item like effect.
Rare/very rare weapon types include: frostbrand, flame tongue, and sunblade, but only the sunblade is specifically a total change in the blade type, rather that bonus damage.
Alternatively, you can use the power of your wizard buddy's wish (or the wish of his simulacrum so he doesn't lose the power to cast it) to allow you to temporarily grant your weapon the properties of a druid's Flame Blade, which more or less duplicates the power of a 2nd level spell.
Method 2: Use the DMG Creating a New Spell rules to import an old spell
There used to be some editions where a priest could temporarily grant the power to cast low level priest spells (in this case, flame blade would be the spell) but I do not believe that spell exists in 5e. However, the DMG has a section on creating spells. You could ask your DM to "create" a spell using the rules on pages 283-284 that creates a druid variant of the 4th level cleric spell Imbue with Spell Ability (which grants a few 1st level spells plus 1 second level spell), and also make it available to your party's Druid.
Then you would simply cast flame blade, which lasts 10 minutes and inflicts 3d6 fire damage, using melee spell attacks.
The problem with this method is the damage is not going to include anything like strength bonuses or enchantments your weapon previously had; it merely fulfills the visual of a Barbarian wielding what appears to be a slashing weapon (scimitar base) that actually deals fire (3d6).
In Hind Sight, if your DM allows you to import spells and then adjust them via the rules on page 283, then you can probably pull an existing spell from somewhere that does exactly what you want, and let your DM pick the spell level.
I would recommend the Simulacrum/Wish for Temporary sunblade-esque or flameblade function for your current weapon. That at least, is close enough to using the Rules as Written to be sorta legal. The only drawback is most people consider anything to do with Wishes to be Rules as Interpreted - but they are still less RAI than importing new spells.
Sim/wish is a pretty hefty order, but you did say level 20, so it's not impossible.