Using http://bgp.he.net/ I know that my AS is AS5089 and I know that this site is hosted on AWS and is in AS14618.
I am also aware that there is one (and only one) BGP route between the 2.
Is there an easy way to determine which ASes this route would pass through, either with (or without!) a Cisco router?
Do a traceroute and check for AS manually using the public IP address. By this way you can form a logical path of ASes.
Cisco routers running BGP will show you the AS number in a traceroute:
R2#traceroute 6.6.6.6
Type escape sequence to abort.
Tracing the route to 6.6.6.6
VRF info: (vrf in name/id, vrf out name/id)
1 192.168.24.4 [AS 400] 9 msec 10 msec 5 msec
2 192.168.46.6 6 msec 7 msec *
If you have a looking glass server close to you then you could try that. Most other looking glass servers support this too btw:
public@route-server.as3257.net-re0> traceroute 8.8.8.8 as-number-lookup
traceroute to 8.8.8.8 (8.8.8.8), 30 hops max, 52 byte packets
1 ae1-150.cr8-fra2.ip4.gtt.net (141.136.111.13) [AS 3257] 0.540 ms 0.525 ms 0.521 ms
2 et-0-0-59.cr11-fra2.ip4.gtt.net (89.149.136.134) [AS 3257] 0.471 ms 0.558 ms 0.476 ms
3 72.14.221.94 (72.14.221.94) [AS 15169] 0.587 ms 0.567 ms 11.879 ms
4 216.239.59.78 (216.239.59.78) [AS 15169] 0.906 ms 1.087 ms 0.874 ms