It is so easy to broadcast your transaction, you don't even need a client 
Just connect to a seed node IP and ask for a list of node IP addresses.
Take one of the addresses and open a socket over port 8333 and use that IP to connect to and say "hi" like this (you may want to change the IP in it but it doesn't matter because it is flexible):
Now that the greeting is over you can start talking with that node. In your case you just wanna tell them about your transaction, so just do that:
Rejected? Try another IP.
This, although being correct steps, was posted just for fun. I may have made mistakes since I didn't use a code, I just made them by hand. If you are interested I can explain with more details.
Just connect to a seed node IP and ask for a list of node IP addresses.
Take one of the addresses and open a socket over port 8333 and use that IP to connect to and say "hi" like this (you may want to change the IP in it but it doesn't matter because it is flexible):
Code:
f9beb4d976657273696f6e0000000000660000008a94bbe97f1101000d00000000000000131b805900000000010000000000000000000000000000000000ffff05ef60c22acf0d0000000000000000000000000000000000ffff23a7f609208df65ca61c4a15f201102f5361746f7368693a302e31342e312f374d070001
Then wait to hear "hi" and a "handshake" back. Then "shake" the hand like this:Code:
f9beb4d976657261636b000000000000000000005df6e0e2
Now that the greeting is over you can start talking with that node. In your case you just wanna tell them about your transaction, so just do that:
Code:
f9beb4d9747800000000000000000000bd000000eae844bb0100000001d9c9f7d12419848477506c8ca09b47f55d6dd4ce8442a467b3cc78596405af211c0000006b483045022100e4feb502f883057194cb0f3dff802177036b52714c94852b6e6f0656a2adbe830220381b45ef1cf7c4937981c3a534364f7b0e7fa7e70de9cae2bc35b62104833043012102d2f573c371c17ae32225044fe2f7582a5ed5ae5835fe0408228f0149ee9a6a12ffffffff01a0860100000000001600146ca8c500e1894e369350c9b0f4f436adf007869500000000
Rejected? Try another IP.
This, although being correct steps, was posted just for fun. I may have made mistakes since I didn't use a code, I just made them by hand. If you are interested I can explain with more details.
I do have a node already running (not synced but still). It's an issue with broadcasting somewhow.
I used sendrawtransaction Hex_Signed_TX and it said there was a scriptpubkey error.
I'm not entirely sure if that code actually works but it looks quite funny anyway
.