Italic nibs exist and on elegant pens.

Where are you located? That might help point you towards a realistic path. Honestly, felt tip is simply not the way to go.

 

As a lefty, fountain pens don't work because of the way the tip drives into the paper and smudging.

 

Not gonna lie, even if you don't want to swap nibs on a standard fountain pen for an italic chisel nib like @jar suggested, there is the poor man's option that we used ALL the time in India when we broke our fountain pens accidentally dropping them on the nib.

1. Snip the nib end to the appropriate chisel width.
2. Gently sand the end smooth with a small bit of sandpaper or a file
3. Finish it smooth with a fine file
4. Go to town writing

 

These are inexpensive cartridge calligraphy pens. Keeping in mind that the only part of the pen most people will ever see is the metal clip, they would be fine riding around on a shirt or jacket pocket - dark, conservative colors. Most felt tip pens have a brighter barrel. And, they’re very reasonably priced on Amazon.

There have to be a wide variety of higher-end fountain pens available with an italic or calligraphy nib too; I prefer a stub nib for thicker cursive, use a Stipula pen with a gold stub nib that I really enjoy (The red one below). Whether the calligraphy nibs on higher-end fountain pens will suit your needs is up to you, but it would be worth visiting a store that specializes in fountain pens and having a conversation if you’re looking for something nicer (eg, more decorative celluloid or acrylic resin barrels, decorated gold nibs, a more decorative clip, etc.) - the dark green one is celluloid, the red is acrylic. The red one has a stub nib; i know that style is also sold with calligraphy nibs, just not sure if the specifications of those nibs would work for you…but to give you a sense, i saw one like the red (with a steel nib, not gold) for under $100 on Amazon.