Don't ignore the responses. If you abuse it too much there is a chance that the api will just block you permanently and is generally seen as not very nice, it does take resources on both ends to process even that response.
The ratelimit crate is an OK solution for this and simple enough to implement/include in your code but can create a miss-match between your code and the API. If they ever change the limits you will need to adjust your program.
A proxy solution seems overly complex in terms of infra to setup and maintain so I would avoid that.
A better solution can depend on the API. Quite often they send back the request quotas you have left either on every request or when you exceed the rate limit. You can build into your client for the API (or create a wrapper if you have not done so already) that understands these values and backs off when the limits are reached or nearly reached.
Otherwise there are various things you can do depending on the complexity rate limit rules they have. The ratelimit crate is probably good for more complex things but you can just delay all requests for a while if the rate-limiting on the API is quite simple.
You can also do an exponential backoff algorithm if you are not sure at all what the rules are (basically quickly retry with an exponentially increasing delay until you get a successful response with an upper limit on the delay). This is also a great all round solution for other types of failures to stop your systems from hammering them if they ever encounter a different problem or go down for some reason. Though not the best if you have more info about the time you should be waiting.