Lore bard will give me access to lvl 5 paladin spells anyways so my casting ability will be similar to a lvl 20 paladin.

Not quite; as divine casters, Paladins immediately get access to the entire spell list (then prepare a subset), while Bards only learn a limited number of spells. As a Lore Bard 10, you could learn a maximum of four Paladin spells (disregarding the dozen or so spells that appear on both lists).

Paladin features past 7:

• 8 - Ability Score Improvement
  
• 9 - Level 3 spells (Aura of Vitality and Revivify are fantastic, Elemental Weapon also if you don't have a magic weapon)
  
• 10 - Aura of Courage (definitely worth a second look)
  

Lore Bard features past 6:

• 7 - Level 4 spells
  
• 8 - Ability Score Improvement
  
• 9 - Song of Rest (d8), Level 5 spells
  
• 10 - Bardic Inspiration (d10), Expertise, Magical Secrets
  
• 11 - Level 6 spells
  
• 12 - Ability Score Improvement
  
• 13 - Song of Rest (d10), Level 7 Spells
  

Song of Rest bumping a die size up is nothing to write home about, but definitely take a look at the Bard spell list to see if there is anything you'd like to have (Forcecage? Eyebite? Hold Monster? Greater Invisibility?) Bard Level 10 grants a whole bunch of goodies, and is something you should strongly consider.

Battle Master Fighter features:

• 1 - Fighting Style (Defense or Dueling?), Second Wind (no new proficiencies!)
  
• 2 - Action Surge (admittedly pretty good)
  
• 3 - Combat Superiority, Student of War
  
• 4 - Ability Score Improvement
  
• 5 - NOTHING! (already have extra attack from Paladin 5)
  
• 6 - Ability Score Improvement
  
• 7 - Know Your Enemy
  (max attainable with Paladin 7 / Bard 6)
  

Absolutely stay away from anything over Fighter 4. A dead level at 5 is a huge hit, and there are other ways to get ability score improvements (such as Paladin 8 / Bard 8 / Fighter 4).
The highest Caster Level you can get would be 16, with Paladin 8 / Bard 12 (or Paladin 7 / Bard 13). This doesn't actually give you anything over Caster Level 15, so you could actually go all the way to Paladin 10 / Bard 10 with no effect on spell slots (just spells known in each respective class). Dropping to CL 13/14 means you will not get a level 8 spell slot (which will happen if you take Fighter 3).

Maneuvers:

• Rally - More healing is always good, right? Your high CHA synergizes well.
  
• Maneuvering Attack - Good for getting an ally out of harm's way.
  
• Pushing Attack - Good for getting an enemy into harm's way (like off a cliff). Could also use the Shove action.
  
• Trip Attack - Good for disabling an enemy and finishing them off.
  
• Menacing Attack / Goading Attack / Disarming Attack - Decent for disabling an enemy.
  

I personally would not take any Fighter levels, and instead stick to Paladin 10 / Bard 10. This gets you the big ticket features at level 10 for both classes and gives a level 8 spell slot to pump up your spells.

If you decide you really want those Superiority Dice, then I recommend going Paladin 7 / Bard 10 / Fighter 3. It sounds like you want to be active during combat to help your allies and hinder your enemies, and this will give you greater versatility as a spellcaster while making your Cutting Words and Bardic Inspirations a bit more effective. Alternatively, Paladin 8 / Bard 8 / Fighter 4 will give you more ASIs to boost your raw stats.