I really don't see any problem with paper wallets, hot/cold wallet on a phone, a seedphrase on a USD stick. I'm not sure where you're from, and I never seen this happening, so I really can't imagine them going through your phone apps. Is that even legal? Also, is this even something you have to do? I mean declaring your bitcoin holdings regardless of the amount?
Well, in the EU you must declare everything over 10.000€.
I would say that in general it is not legal for them to go though your apps but I have seen border control programs where it does seem mandatory, or at least they can make it more difficult for you to enter the country or hold you if you don't cooperate.
I was thinking hypothetically.
You are right about that they will make it difficult for you to enter the country if you don't cooperate because they must make sure everything is safe both what is coming to the country and what is going out of the country but if we take look at it very well, checking on someone app is not their duty, I don't hold something valuable of BTC in your app is illegal unless the agent wants to collect a bribe with you and start doing some unnecessary searching, nothing can be hidden in the wallet if not a coin, any agent who asks for your phone and wants to check your wallet is not an agent, he is a criminal or he is a thief because I don't know what they looking for in someone private saved, but it depends on the country you are living or the kind of agents the rules guide your country.
The best way to be safe with the agent even when they enter your phone and would not find anything is to save your seed phrase somewhere else,
if you have someone you trust like your dad or mom you can write it and send it to them as a message and also send your coin to someone who you trust after you have passed the agent checkpoints or getting to another country the person can send you your coin and you can also go through the message you send to who you trust and get your seed phrase to be able to have access to your wallet without any stress.
I am having trouble seeing what advantage you have in regards to sending your seed to someone else, including your seeming presumption that there would be any secure way to send such message to someone else, just like phillipma's encoded message that he was describing from earlier? There could be complicated ways to send seedphrases or passwords, and perhaps you are able to get access to funds, but then if anyone ever figures out or sees such system in the future (maybe past messages that you sent that are figured out in the future), then if you had already moved your coins to new addresses, then that might not be a problem to have some temporary vulnerabilities that are not very likely to get exploited, as long as you don't overly complicated your system too much that you cannot recall how to put all the pieces back together.
I do like the idea of keeping some form of back up recovery at country A while traveling to country B in the event that somehow your system fails in regards to getting to country B, and even I am not assuming traveling just on a trip but wanting to take your whole life savings with you, yet if you have your back up recovery in Country A, and once you make it to your country B destination, you might have to figure out how to destroy your back up recovery in country A so that no one comes in contact with it.. unless again you are transferring to a new wallet.. .which sometimes may be prudent to create new wallets and transfer, even though frequently I have considered both vulnerabilities in creating new wallets and then transferring coins, which could be a bit cumbersome to carry out.
One of the issues that I had considered with the hidden passphrase is that if your device gets confiscated and security is able to get your seed off of it, yet if there is a bit of time, you can move the coins prior to their breaking into it, if you are not detained, yet if you are detained then they have more time to break into your device... various forms of encryption and/or secure elements might be helpful in those regards to the extent that some of us are sufficiently comfortable with the employment of such systems.. or even needing to learn about such encryption systems.