It would definitely be sweet if your DM gave you a mithril chain shirt that worked like light armor. No idea why the chain shirt didn't stay light armor.
A general overview:
The chain shirt was light armor in 3rd Edition, granting +1 AC above studded leather, which in 5E terms would be expressed as AC 13 + Dex modifier (not 14).
The 5E elven chain is a +1 chain shirt that does not require medium armor proficiency. The most straight-forward interpretation makes it still a medium armor, making it not stackable with bladesinging.
5E mithral can't be applied to light armors. Its general benefit to waive the Strength/Speed and Stealth disadvantages. For chain shirts and breastplates (which do not have these restrictions) the benefit is instead "can be worn under normal clothes" which is more of a roleplaying benefit since there is no bonus to any relevant combat parameter.
Notable is that during the playtest, a "mithral shirt" was explicitly in the "PHB" (the equipment chapter of the playtest) being a light armor granting the AC 13 + Dex of a chain shirt equivalence costing 5000 gp.
(At that time all three armor categories had expensive but not magical armor to strive for, such as "dragon leather" or "mithral scale". Note how regular plate mail was priced as the equivalent 5000 gp heavy armor - a remnant of this can be seen in how plate mail is in 5E "semi special": it isn't magical, but no starting character can afford plate mail straight away)
Of course, during the playtest studded leather was considered medium armor, so my best guess is that the designers didn't feel the "old" armor types were sacred, and switched them around to test various configurations, and that the main reason we ended up with medium chain shirt is more or less by chance - simply that this was the current configuration when the playtest froze that particular aspect.
A "a mithril chain shirt that worked like light armor" is the equivalent of a +1 studded leather (that you can wear under regular clothing) if I assume by "work like light armor" you mean not only "counts as light armor for purposes of bladesinging" but also "no max Dex modifier", so I guess if your DM is okay with a +1 studded leather she should be okay with this too.