The eldritch knight just doesn't provide enough magic for what I want, the bladesinger leans too much to the magic side of things.
I think you should tailor your feats to address these deficiencies.
1. For the EK pick up Fey touched, Shadow touched, magic initiate, artificer initiate, eldritch adept taking a spell invocation and metamagic adept. By 14th level That will give you 3 extra cantrips, 7 more known spells, 5 more 1st level castings, 2 more 2nd level castings and metamagic with 2 sorcery points.
At 14th level you have 6 cantrips, 17 spells know and equivalent of 16 spell slots plus 2 sorcery points. This compares to a full blown sorcerer who has 6 cantrips, 13 spells and 17 slots. This character has a ton of magic, what he doesn't have is high-level spells.
2. For the bladesinger option get weapon master, crossbow expert and sharpshooter and they can do a ton of damage with a hand crossbow. With a custom lineage, taking weapon master at first level and prioitzing dexterity they can have a 20Dex and these three feats by 12th level. They will rock 3d6+45 DPR while sporting a 20AC in mage armor and bladesong with shield to boost it to 25.
You can melee this character with crossbow expert, but if you want to actually melee, with melee weapons you can take weapon master at 1st, ASIs at 4/8/12 and at 12th level runing a upcast shadowblade and a short sword you can do 8d8+1d6+10 while sporting a 21AC in bladesong with shield for 26.
3. Third option which would work well is a multiclass artificer-EK or Artificer-Bladesinger. Go to level 5 in battlesmith then go the rest of the way as an EK or BS. Go with the eldritch knight if you want to use heavy weapons. Either build dump strength though take 18 Intelligence at 1st level with an int boosting feat, 14 Dexterity to allow ftr multiclass. Take an ASI for 20int, then take weapon feats (GWM, Sentinel, Pole Arm Master, sharpshooter, piercer, slasher ....). At 3rd with Battle Ready you can use intelligence for ANY magic weapon attack and you can infuse a magic weapon every morning (2 total), meaning when you get +5 for both strength weapons and ranged weapons, while only having an 8 strength and 14 dex. You can build to both GWM and Sharpshooter if you want. Of the two, the fighter is a more versatile fighter, if you go bladesinger you will need to use a 1-handed or versatile weapon in bladesong, but it brings all of the weapons to the table - long swords, morning stars, whatever you want. You have proficiency in all of them and you don't need it to be finesse because you are attacking with intelligence.
