Your Huawei E585 shouldn't be the problem as it supports RX diversity for all the UMTS frequency bands, it supports (there are two versions: 850/1900/2100 and 900/1900/2100).
Assuming that you have the American GSM version of the iPhone 5 (model A1428 ) it will indeed not work on any European LTE network, as it supports only LTE band 4 (1700/2100 MHz aka AWS) and 17 (700 MHz), of which none will be used in Europe.
The European iPhone 5 (model A1429) supports LTE bands 1 (2100 MHz), 3 (1800 MHz) and 5 (850MHz), which is of limited use, too, as much of European LTE coverage will be based on LTE band 20 (800 MHz), which covers the spectrum formerly used for terrestrial TV broadcasting and is also know as "digital dividend". Those LTE frequency bands supported by the European iPhone 5 will only be used by some operators who will use them for urban coverage.
As of checking credit you can do that through the customer portal or by USSD (dial *130#), however USSD is quite complicated to use on a mifi (you need to run your Huawei E585 attached to a computer by USB in modem mode and use an USSD-tool like
MDMA or use a serial console to execute rather complicated AT commands). If you cannot register on the customer portal you will need to put your swisscom SIM in another phone to submit the USSD code.
That offer for CHF 36 is called "swisscom pocket connect" and indeed includes a Mifi and 5GB of data valid for 5 days. You will also need to deposit CHF 129 and prolonging the rental costs CHF 4 per day but you need to prolong by at least 3 days:
VILLAGES | Jungfrau Region | Grindelwald | Wengen | Mrren | Lauterbrunnen
Pocket Connect - Mobiler Hotspot - Swisscom - Internet-Angebot für Gäste - Surfen - Privatkunden - Swisscom
That's not very cheap in my eyes.