Have a look at Rick Wilson’s composite C19th chart for a list of variant 3rd 8ve fingerings. Not only can you experiment with which offers the best in tune one on your flute, you can also see which combinations of C nat and E fingerings offer you the easiest and most reliable transition.
FWIW, the most in tune 3rd 8ve E on many Rudall style flutes is xxo, xxx, (both G# and Eb keys open) but that is an awkward one to combine with anything much else. The “standard” fingerings are xxo xxo, (sharp) and xxo oxx, (flat). On Pratten style flutes the keyed high C nat (x,oo ooo,)is usually quite good (and should transit legato well enough to an E), but is rarely so on Rudall types (often flat) and you need to use xox xx,o,. If it is any consolation, high E is the worst note on the simple system flute.
I think (just been experimenting myself) that with a bit of practice to get the finger changes really clean and your embouchure (adjustment) just right, it is perfectly doable. The xox xx,o, C nat to xxo xxo, E should work well enough, but you could see how your flute (and you) behave with oxx xxo(,) for C going to either xxo xxo, or xxo oxx, for E, or oxo xxo/x(,) for C (try with/without R3 and ditto Eb key) to those two E options. If the keyed C nat is acceptably in tune on your flute, even going to the awkward-but-best E (x,oo ooo, to xxo, xxx,) is not too bad a switch. There are probably other variants, but that covers the main ones. Virtually all of the possible fingerings need the Eb key open. I suspect you wouldn’t need the G# in the fingering you quote if you opened the Eb instead. I just tried your fingering (xxo, xxo) and it does work on the Fentum flute I have in hand, but is similarly flat as xxo oxx, whereas xxo, xxo, is, as one might expect, sharp unless you close R3. But NB, these thing vary from flute to flute!!!
Once you have found the best combination, you will still have to work on both the accurate embouchure adjustment and a very clean finger-switch - any lack of synchronicity in getting the fingers accurately onto their holes/keys will make you miss the target note and get a harmonic, wrong note or even nothing at all but hiss!
Final point on a different tack; I’m no period performance expert, but I believe that in best Baroque technique almost all notes are articulated in some way, virtually none are truly legato/slurred, though the articulation may be ever so gentle and almost more a link than a separation - perhaps an r or l or very soft d articulation might achieve the effect you seek?
Good luck!