packets wouldn't alternate like that, it would be streams that alternate ... To do proper load balancing, you have to have a common IP address (or block) that is routed by both ISP's. You'd then be making routing decisions on routing metrics, and the routing metrics wouldn't change all that much (and shouldn't for your network and others to be considered "stable")
Now, doing improper load balancing with a multi-homed router (wan modem) would still keep streams with respective paths until one ISP fails, and then you have to hope the application you're running is tolerant of source ip changes. Which I don't believe dish/sling is.
you start streaming from ISP1, you're at your office watching some live TV, ISP1 goes down, ISP2 takes over, Sling Web disconnects because its no longer getting communication from ISP1 but.. wait.. now its seeing ISP2 ... Dish classifies it a Failure ... packet sequence from incorrect source .. User you have Error 36 (I think that's the number ... that 33 or 13 maybe?) the device was connected, but now the connection has been broken (and dish's useless drones would tell you to reboot the dish receiver).
You'd wait for the receiver to reach out again, tell Dish/Sling "hey I'm User PATECO's dish reciver" dish records the new IP confirms the connection, you the User then refresh/reconnect and streaming resumes.
The time for that kind of break in communication minutes to hours ... it all depends on how quickly your receiver re-establishes itself from the new IP (re-establishes by phoning home to papa dish) You're not at home, so you can't go power cycle the dish, and oh, since the path dish knows about doesn't work any more, you can't reboot from on line web pages either.. not until dish has gotten the new ip your receiver is behind..
ya follow? That's why you want your own set of IP Addresses ... and both ISP's routing them.. and then the routing decisions are based on best path by your router's choice ... If you are using your own set of IP's ... then it doesn't matter how your receiver gets to the Dish servers.. because it is the same Source IP... Yours ...