Network protocols are designed to avoid problems with "endian-ness". IBM dominated the computer arena up until about the mid 1980s, and it was "big-ended" (MSB first). As far as I know, the other "seven dwarfs" of the industry were likewise big-ended.
It was DEC that made the little-end approach popular. Why they did it, I have no idea - probably made manufacturing cheaper to a more cost-sensitive market - but when Intel started making microprocessors, they did likewise.
I don't care for little-end myself. It's a bytewise-discontinuous mode, which wreaks havoc on memory-to-memory bitwise operations and comes in at least 2 sub-flavors when you start talking 4-byte words and larger, since you can be byte-swapped within 16-bit parts and "big-ended" for the pairs, OR you can be both byte-swapped AND word-swapped.
But it's not going away anytime soon.
However, dealing with this mess is one of the reasons why a lot of network protocols are formally specified in ASN.1 notation, which deals with that sort of stuff as a core principle.
If you're getting connections bounced or closed, however, I doubt it's the data format. The parts of the data packets that that layer of the network stack deals with are supplied by the hardware and OS, which already know what order the bits should be in and deal accordingly.
More likely you have a networking problem. Could be bad cabling, flakey router/switch, traffic overload, or a problem with the destination machine (mis-configured networking or application responds too slow).