I have learned that fratineto and frateto are used for little sister and little brother. What about big sister and big brother? Fratinego and Fratego sounds strange to me.
I disagree that frat(in)eto is the only or even the best way to say "little brother/sister." What I say is (mal)granda frat(in)o.
A quick check in the Tekstaro (checking only "Zamenhofaj" texts) shows that forms with granda are more common, and it seems that forms with -et- usually refer to younger siblings who are still small, or it's used as a caress form.
Checking the Tekstaro for adjectives used immediately before "frat-" suggests a few good aleternatives.
- pli juna fratino
- maljuna fratino
- la malgrandan fratinon
- ia malgranda fratino
- viajn plej maljunajn kaj plej junajn fratinojn;
- La plej aĝa frato
- liaj du pli aĝaj fratoj.
- mia malgranda fratino
- mia malgranda frateto
My recomendation - any variation with aĝa or granda is fine. Juna works too.
It is not routine for the concept to be compressed into one word; people generally say pli aĝa and malpli aĝa (pli juna) frat(in)o.
However, -eto and -ego do appear in print and they seem perfectly fine to me, a very sensible and appropriate use of those suffixes, particularly if relative age is socially important (e.g. in East Asia). So if you are hunting for a single word then feel free to say fratego, fratinego, etc.