The .380 is a poor choice for a number of reasons related to the action. First off, all the .380 barrels are fixed, which is good. Second, all the .380s are simple blowback weapons, which is bad.

The suppressor will increase bore pressure a bit, resulting in faster opening of the action (allowing more gas/noise to escape through the ejection port). In addition, the faster slide velocity makes the weapon cycle faster, and the magazine spring too often does not get the next cartridge in position for feeding in time. This results in a stovepipe jam.

Since you are considering .380, terminal ballistics does not seem to be that much of an issue. In this case, I would suggest .22LR for a first suppressor. It is significantly smaller, lighter, cheaper, and has better performance (having to deal with less gas). It is also more fun and seriously more economical shooting.

True, but it is possible to delay action opening by machining a shallow ring in the chamber. Makes easier for the forensic folks to identify the brass left at the crime scene.