Can I replace SHA-1 with SHA-512/160 to address Shambles? The destination is software (within a remote trust boundary) that expects SHA-1 results. Would it be safer for the source (that's within my own trust boundary) to replace my SHA-1 computation at my source with a SHA-512/160 instead?
Generic collisions

The generic collision attack on SHA-512 trimmed to $n=160$-bit will require $2^{80}$ complexity by the birthday paradox with a 50% success probability. The generic attack doesn't require any knowledge about the internals of the target hash function. It is about collecting hash outputs and looking collision among them by building a table and hoping that you find a collision. The cost is $O(2^{n/2})$ time and $O(2^{n/2})$ space to store the table.  

If you are using a variant of Pollard's $\rho$ you can achieve constant $O(1)$-space with $O(2^{n/2})$-time. The best known is the van Oorschot–Wiener Time-Momory Trade-off technique with cost $O(2^{n/2})$. This can be parallelized with a linear speedup. Use $O(2^{n/4})$ machine to parallelize and get an answer in $O(2^{n/4})$ time at the same cost of $O(2^{n/2})$. 

If you consider the Quantum attacks on hash functions, you should use 384 or 512 output sizes due to Brassard et. al' work. That has $\mathcal{O}(\sqrt[3]{n})$ attack time for n-bit hash function (for 256-bit $\mathcal{O}(\sqrt[3]{2^{256}}) \approx \mathcal{O}(2^{85})$). See the timing table in this answer.

SHA-1 weakness and Shamble

The researches use weaknesses that lie in the internals of SHA-1 and The Shambles' researches reduced the collision to $2^{61.2}$, see the list of other work in here. The Shambles' team used 900 GPUs that cost around $75.000 with their new method and calculated the forgery at most in two months. This is why they called practical. Their new records;


They reduced the use of neutral bits BCJ+05 and boomerangs JP07 from $2^{64.7}$ to $2^{61.2}$ 
Also, they improved graph-based technique (LP19) to compute Chosen-Prefix Collision from $2^{67.1}$ to $2^{63.4}$.


The Machines

There are machines already exist way before the attack's 900 GPUs, like the Summit in Oak Ridge that can reach $2^{63}$ SHA-1 calculations in a day. As long as, one cannot find a weakness in SHA-512/160, it will have an 80-bit classical collision resistance. 

80-bit is no longer secure

Today, 80-bit collision resistance for any hash function is not considered secure. The NIST removed SHA-1 signatures from the recommendation in 2011. 


  NIST formally deprecated use of SHA-1 in 2011 [NISTSP800-131A-R2] and disallowed its use for digital signatures at the end of 2013.


R2 in 2019 on page 18 NIST allows it for nonsignature, although by that time it's clearly a bad idea (Thanks to Dave).

Actually, one can consider that it is not the SHA-1 is removed, the 160-bit hash output is removed from the recommendation. The 160-bit was related to 80-bit security that was enough during the early 2000s. And, note that the 50% probability of the birthday attack is not negligible in the view of the attackers. One must consider the lower probabilities.

Recommendation

It is wiser to update the software to new recommendations like the SHA3 series with good hash sizes like 256, 384, or 512 if you are working with the US government. If not, you can also use other good hash functions like Blake2. If you consider Quantum attacks, you should bigger than 256.

You can use it if you are satisfied but not advised.

Migrating the software shouldn't be a big problem. It is strongly recommended.