I have never played this multiclass, but I never hang back with a single class bladesinger. If you are doing that you are playing the subclass wrong IMO and your AC needs to be above 22 with shield and bladesong active at low levels to do it well. Having fewer spell slots for shield might change that a bit though and this character will be a little softer than a single-class mage.At 10th level you'll be kind of a subpar melee fighter, who should be hanging back and casting spells. Prior to that you'll be an awful melee fighter who should be hanging back and casting spells.
I would have thought a melee fighting tank would be a better option in a party with 2-3 ranged PCs.
Dex based Elven (+2 Dex) Eldritch Knight using a Double bladed Scimitar looks pretty good with those stats, taking the Revenant blade feat at 4th level (which also raises your Dex to 20 as an Elf), and then taking GWM at 6th, and whatever else you want at 8th and 10th.
Your AC should be fine with +1 from the Revenant blade feat, a high Dex (or heavy armour) and Defence style (AC 20) with Shield online from 3rd.
If your DM considers Custom lineage [Elf] to qualify for Elven feats, take Revenant Blade at 1st level for a Starting Dex of 20 (plus a finessable and heavy double weapon).
Regarding damage I think this is better than you might think with this multiclass - assuming you land sneak attack, damage without shadow blade at 10th level is 30.5DPR to one enemy and 7.5DPR to another within 5 feet: 2d8(rapier twice) + 1d8(GFB) + 2d6(SA) + 10(Dex) plus another 1d8+3(GFB) to the second enemy. 38 points a turn is not that bad and that is without using any limited use abilities or a -5 to hit. If you use shadow blade that would be 35 against one enemy or 44 if you use a 3rd/4th level slot. There are lots of other variations too; shadow blade with short sword GFB - 34 to one enemy, 7.5 to a second. With shadow blade there is another 3.5 on the table for a BA offhand attack too (although there are lots of other BA options).
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