[RPG] How to take multiple fighting styles

No, you can't combine them in this way.

The key point is in the description of the Two-Weapon Fighting rule (PHB, p. 195):

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand.

The relevant issue is this: when you take the Attack action, is there a weapon in your other hand? If there is, you can't get the benefit of Dueling. If there is not, you can't make an off-hand attack using Two-Weapon Fighting.

(Note the present tense "that you're holding in the other hand", not "that you will be holding later in the turn" or "that you were holding earlier".)

The answer by @Wizo is correct, as is your own question's assessment of your question--as per 5e RAW you cannot gain a new fighting style through any means except multiclassing.

My suggestion would be to dip into fighter for one level to grab your second fighting style--but there are considerations.

1. Multiclassing delays class abilities - It is often recommended that a base class take at least 5 levels in that base class to start with so that they gain a second attack and/or 3rd level spell (Ranger's case is 2nd attack). So, if you do multiclass, waiting until level 5 is a good tip, although it could be more contextually dependent than that.

2. Ability score improvements are granted by class level - Unlike previous editions and Pathfinder, ability score improvements are granted by class level, not character level. So, multiclassing into more than one class will result in suboptimal ability scores at higher character levels. It may be more optimal for you to simply focus on Dexterity and use finesse weapons--you may not have the

3. No capstone abilities - The second real drawback to multiclassing, which doesn't often come into play (because many tables don't rise up to level 20...) is that levels are capped at 20, meaning that characters who multiclass won't ever get their capstone abilities, i.e. you can gain no more class levels and thus you can not gain any more class features.

4. Two Weapon Fighting is only... ok - In this player's opinion, dual-wielding is only good at low levels but loses effectiveness quickly. The major problem with dual wielding is that it takes a bonus action, and as you level up your class will have more and more bonus actions to use, meaning that at low levels it's an additional attack but at later levels it becomes just another option. I'd suggest Defense or Dueling.

All that said, I totally understand the flavour of a Ranger requiring something more than just a ranged attack--TWF is very cool and flavourful, but using a finesse weapon in a single hand makes Dueling a better choice for the long run. However, in this player's mind neither may be worth the dip into Fighter.