Ultrasonic signal detection
These are not the codes you are looking for...

As I mentioned in the comments, there are quite a number of ways to do robust TDOA. (Cross-correlation with Linear Chirps, Exponential Chirps, and CDMA-type methods). You have already built a TDOA system utilizing codes, (and that is indeed a good choice over linear-chirps if you need robustness to doppler), however you are limiting yourself artificially in two ways:


Barker codes only go up to length $13$. We can however make PN-sequence codes of arbitrary length to get a lot more coding gain.
The use of only $1$ bit in your transmission. We can encode an entire preamble of many bits to transmit, gaining further resilience to multipath.


Use a PN-Sequence: 

Thus, very simply, change the codes you use to modulate your carrier by: Use PN-Sequences instead. PN generated codes can be of (nearly) arbitrary length, and can be generated via LFSRs. (They also go by the name 'whiteners' in some texts). Here are three PN-sequences of length $31$, $61$, and $127$ respectively. 

PN_31 = [ 1  1 -1 -1  1  1 -1  1 -1 -1  1 -1 -1 -1 -1  1 -1  1 -1  1  1  1 -1  1  1 -1 -1 -1  1  1  1];

PN_61 = [ 1  1  1 -1  1  1 -1  1 -1 -1  1 -1 -1  1  1  1 -1 -1 -1  1 -1  1  1  1  1 -1 -1  1 ...
     -1  1 -1 -1 -1  1  1 -1 -1 -1 -1  1 -1 -1 -1 -1 -1  1  1  1  1  1  1 -1  1 -1  1 -1 ...
      1  1 -1 -1  1  1 -1];

PN_127 = [-1     1     1     1    -1     1    -1    -1     1    -1     1     1    -1    -1    -1     1     1    -1     1     1     1     1    -1     1     1    -1     1    -1 ...
       1     1    -1     1     1    -1    -1     1    -1    -1     1    -1    -1    -1     1     1     1    -1    -1    -1    -1     1    -1     1     1     1     1     1 ...
      -1    -1     1    -1     1    -1     1     1     1    -1    -1     1     1    -1     1    -1    -1    -1     1    -1    -1     1     1     1     1    -1    -1    -1 ...
       1    -1     1    -1    -1    -1    -1     1     1    -1    -1    -1    -1    -1     1    -1    -1    -1    -1    -1    -1     1     1     1     1     1     1     1 ...
      -1     1    -1     1    -1     1    -1    -1     1     1    -1    -1     1     1     1];


The circular and linear auto-correlations of the sequences are shown below. They will clearly yield white spectra, but more than that, we are no longer limited to $13$ chip lengths. In fact, the last code, PN_127, yields a coding gain of $10 \ log [\frac{127}{13} ] \approx 10$ dB gain over the barker sequence, all the while guaranteeing white spectra. 



Transmit a preamble:

In your particular application, you mentioned that you were only transmitting one bit. You should try to avoid this if you can help it, and transmit as many bits as your application can allow, to get further coding gain out. 

This is what is commonly done on communication protocols to align with the beginning of a packet. A (known) preamble is transmitted, composed of many bits. Each bit, is composed of many chips. (In our example, $31$, $61$, or $127$ chips with either of the above PN codes). Lastly, the bit sequence itself can be composed of yet another PN sequence, or if you like, you may transmit $13$ bits composing a barker pattern, with each bit being composed of either one of the the above PN sequences. 



Try one or both of those solutions, and put up your results. I expect there to be tangible improvements that we can then iterate on. (Pulse shaping, different/longer PN sequences, etc).