I think there is no faster way than spending three levels in multi-classing unless you can get DM buy-in or use a magic item like the helm of telepathy.
The simplest solution might be to get a helm of telepathy, it is an uncommon item, so it might be possible to get this even in a relatively low-magic campaign, in a major metropolis.
Barring that, the various feats that let you pick a spell (like fey touched or magic initiate) only give you first level spells as your pick, so they won't work.
The shortest route I can think of how you can do this innately is picking up three levels in a class like sorcerer or warlock, that has the spell on their list.
If you want to go that way, warlock might be a good option for you as a fighter, as you can pick a hexblade which can reinforce your prowess as a melee fighter with spells like shield, wrathful smite, the hexblade's curse and access to hex all from level one, so you are not wasting two levels just to bridge towards suggestion, without getting something out of them that works for what you already do. Sorcerer might also work; Wizard is probably not as good given your low intelligence score.
As DragonOfFlame's answer states, you might also be able to change your subclass retroactively, and get there within two levels, again this only works with your DM's approval.
Lastly, in addition to published rules, if your DM is willing to work with you, you could work out a feat with them that allows you to cast the spell 2-3 times a day, for example Charisma bonus times could work. The power value of being able to cast Suggestion once a day is worth about 3 points on Detect Balance, while the value of a full feat or Ability Score Improvement is 8 points. Of course, such homebrew is not an official way.