Is it possible to get the below behaviour with InversifyJS:

constructor(IDependency resolvedDependency, string myLiteral)
                        ^                          ^
                Automatically resolve      Defined Literal

If so, what's the best way to go about it?

There are a few things that you can do. Here I will post a few of them...

Injecting the literal as a constant value

let TYPES = {
    IWarrior: Symbol("IWarrior"),
    IWeapon: Symbol("IWeapon"),
    rank: Symbol("rank")
}

interface IWeapon {}

@injectable()
class Katana implements IWeapon {}

interface IWarrior {
    weapon: IWeapon;
    rank: string;
}

@injectable()
class Warrior implements IWarrior {
    public weapon: IWeapon;
    public rank: string;
    public constructor(
        @inject(TYPES.IWeapon) weapon: IWeapon,
        @inject(TYPES.rank) rank: string
    ) {
        this.weapon = weapon;
        this.rank = rank;
    }
}

let kernel = new Kernel();
kernel.bind<IWarrior>(TYPES.IWarrior).to(Warrior);
kernel.bind<IWeapon>(TYPES.IWeapon).to(Katana);
kernel.bind<string>(TYPES.rank).toConstantValue("master");

Injecting the literal based on the context

Using contextual constraints you could inject a constant value for a particular context:

let kernel = new Kernel();
kernel.bind<IWarrior>(TYPES.IWarrior).to(Warrior);
kernel.bind<IWeapon>(TYPES.IWeapon).to(Katana);

kernel.bind<string>(TYPES.rank)
    .toConstantValue("master")
    .whenTargetTagged("rank", "master");

kernel.bind<string>(TYPES.rank)
      .toConstantValue("student")
      .whenTargetTagged("rank", "student");

let master = kernel.getTagged(TYPES.IWarrior, "rank", "master");
let student = kernel.getTagged(TYPES.IWarrior, "rank", "student");

injecting a factory

let TYPES = {
    IWarrior: Symbol("IWarrior"),
    IWeapon: Symbol("IWeapon"),
    IFactoryOfIWarrior: Symbol("IFactory<IWarrior>")
}

interface IWeapon {}

@injectable()
class Katana implements IWeapon {}

interface IWarrior {
    weapon: IWeapon;
    rank: string;
}

@injectable()
class Warrior implements IWarrior {
    public weapon: IWeapon;
    public rank: string;
    public constructor(
        @inject(TYPES.IWeapon) weapon: IWeapon
    ) {
        this.weapon = weapon;
        this.rank = null; // important!
    }
}

let kernel = new Kernel();
kernel.bind<IWarrior>(TYPES.IWarrior).to(Warrior);
kernel.bind<IWeapon>(TYPES.IWeapon).to(Katana);

kernel.bind<inversify.IFactory<IWarrior>>(TYPES.IFactoryOfIWarrior)
    .toFactory<IWarrior>((context) => {
        return (rank: string) => {
            let warrior = context.kernel.get<IWarrior>(TYPES.IWarrior);
            warrior.rank = rank;
            return warrior;
        };
    });

let warriorFactory = kernel.get<inversify.IFactory<IWarrior>>(TYPES.IFactoryOfIWarrior);

let master = warriorFactory("master");
let student = warriorFactory("student");

You can inject factories into other classes:

@injectable()
class Army {
    private _warriorFactory: (rank: string) => IWarrior;
    private _soldiers: IWarrior[];
    public constructor(
        @inject(TYPES.IFactoryOfIWarrior) warriorFactory: (rank: string) => IWarrior
    ) {
        this._warriorFactory = warriorFactory;
        this._soldiers = [];
    }
    public newRecruit(rank: string) {
        this._soldiers.push(this._warriorFactory(rank));
    }
}

I think that the best solution is to use a factory.