I take it there are no differences in terms of security when using large or small amounts, including 0.001 BTC?
The main security consideration is the same with any platform, service, or exchange, to which you have to deposit your coins; mainly that you do not have custody of your coins for the duration that they are on said platform. With that in mind, you might prefer to split up (for example) 1 BTC in to a number of smaller deposits and withdraw all or some of each deposit before sending the next deposit, so you are never leaving more than, say, 0.2 BTC with ChipMixer at any time. Or you may decide that is more hassle than it is worth and just deposit the full 1 BTC in one go. Up to you.
There are some privacy considerations, though. Understandably, larger chips are withdrawn far less often than smaller chips. If you were to deposit 1.024 BTC (or slightly more), and then immediately withdraw a 1.024 BTC chip, then you weaken your privacy.
In addition to waiting some time between deposits and withdrawals, you can also split and merge deposits to aid with both the points I have made above. You could deposit 0.2 BTC, and then a few days later withdraw 0.128 BTC and leave 0.072 BTC with ChipMixer. Then, a few days after that, you deposit another 0.2 BTC, giving you 0.272 BTC, and allowing you to withdraw 0.256 BTC. By leaving some coins on ChipMixer and depositing and withdrawing from a pool over a period of days, weeks, or months, you maximize your privacy, and you can also make a number of smaller transactions rather than mixing your full stack at once.
Very well explained on how to approach this aspect if you do really want to get the benefits of using a mixing service.
After all, you want to keep your privacy, hence, using this kind of service.
Also, if you will not be using like ChipMixer, better use a mixer service who has active thread in this forum.
In that manner, in case you will encounter any issues, they can help you resolve your situation.
Don't trust services that you think you can't get a hold of their support.
Because as you are aiming to maintain your privacy, you are also increasing the chance of losing your funds.