Here are some thoughts on a third way we can tackle our BIP 70 usability problem sans servers: by finding an upgrade to QR codes that give us more space and then optimising the hell out of BIP70 to make it fit.
Better QR codes
Let's start with this paper, High Capacity Colored Two Dimensional Codes. It develops an upgrade to standard QR codes that extend them with the use of colour. The resulting codes have ~4x the capacity but similar levels of scanning robustness.
It works by overlaying one QR code on top of another using shades of grey. The resulting code is still scannable by older applications (backwards compatibility!) but an enhanced reader can also extract the second code. They explicitly mention digital signatures as a possible use case.
In both cases the code does not appear to be available but the same approach was used: extend libqrcode for creation and ZXing for decoding (Android). We could ask the authors and see if they're willing to open source their work.
BIP 70 has the potential to add many features. But most of them, even the extensions currently proposed only as ideas, can be expressed with relatively few bytes.
So with a 4x boost in capacity, or a 2x boost with backwards compat, what could we do?
Optimised BIP70
If we define our own certificate formats and by implication our own CAs, then we can easily make a certificate be 32 bytes for the ECC signature+length of the asserted textual identity, e.g. email address.
Can we go smaller? Arguably, yes. 32 bytes for a signature is for Really Strong Security™ (a 256 bit curve), which gives 128 bits of security. If we are willing to accept that a strong adversary could eventually forge a certificate, we can drop down to a weaker curve, like a 128 bit cure with 64 bits of security. This is well within reach of, say, an academic team but would still pose a significant hurdle for run of the mill payment fraudsters. If these short CA keys expired frequently, like once a month, the system could still be secure enough.
As we are defining our own PKI we can make CA keys expire however frequently we like, up to the expiry period of the BIP70 request itself. Thus certificates that expire monthly is not an issue if the wallet has a way to automatically refresh the certificate by using a longer term stronger credential that it keeps around on disk.
If we accept a single payment address i.e. no clever tricks around merge avoidance, such a QR code could look like this:
However this requires text mode and wastes bytes at the front for the URI type.
If we're willing to accept QR codes that can't be read by a standalone app and which requires an embedded reader, then we can just scrap the legacy and serialise a binary BIP70 request directly into the QR code. Andreas' wallet, for example, can already handle this because it has an embedded QR reader. I don't know what the situation on iOS is like.
If we were to use the DualCodes system we could define the primary QR code as being an unsigned payment request, and the second layer as being the signature/pki data.
Getting response data back to the recipient
One reason to have a store/forward network is the "forward" part: we don't only want to host a static PaymentRequest, but also receive a private response e.g. for the memo field, or to implement the well known "Stealth Address" / ECDH in the payment protocol proposals:
Stealth addresses try and (ab)use the block chain as a store/forward layer and break SPV in the process as well as wasting lots of resources. ECDH in BIP70 avoids those issues but at the cost of requiring a separate store-and-forward network with some notion of account privacy.
These ideas come with another steep price: restoring a wallet from seed words is no longer possible. You must have the extra random data to calculate the private keys for money sent to you :( If you lose the extra data you lose the money. It can be fixed but only by having wallets regularly sweep the sent money to keys derived from the BIP32 seed, meaning privacy-hurting merging and extra traffic.
I don't know of any way to solve this except by using some servers, somewhere, that store the Payment messages for people: potentially for a long period of time. If we have such servers, then having them host BIP70 requests is not a big extra requirement.
I have imagined this being a p2p-ish network of HTTPS servers that accept POSTs and GETs. But if we are thinking about alternatives, it could also be a separate service of the existing Bitcoin P2P network. That's what OP_RETURN (ab)use effectively does. But as these messages don't really have to be kept forever, a different system could be used: Payment messages could be broadcast along with their transactions and stored at every node, waiting for download. But unlike regular transactions, they are not stored forever in a block chain. They are just written to disk and eventually erased, perhaps, ordered in a mempool like way where more fee attached == stored for longer, even though the nodes storing the data aren't actually receiving the fee.
A signature over the Payment metadata using the same output keys as the transaction would bind them together for the purposes of broadcast, but doesn't need to be stored after that.
As the data storage is just a helpful service but not fundamentally required, nodes could shard themselves by announcing in their addr messages that they only store Payment metadata for e.g. the half which have a hash starting with a one bit. And when outputs are seen being spent, the associated Payment metadata can be erased too, as by then it's fair to assume that the users wallet has downloaded the metadata and no longer cares about it.
Of course you have then all the regular DoS issues. But any P2P network that stores data on the behalf of others has these.