Japanese - Asking to reserve a (train) seat.?

reserve a *train* seat >>> Hmm... if you're talking about buying a train/Shinkansen ticket at "JR 緑の窓口", we normally say like

-すいません。新大阪から東京までの『のぞみ』(の)指定券[一枚。/ お願いします。/ 一枚お願いします。]。

-すいません。東京までの(新幹線の or『ひかり』) 指定券[一枚。/ お願いします。etc.]

-すいません。10時32分の『のぞみ』で、東京まで指定券[一枚。/ お願いします。etc.]

-すいません。((新大阪から)東京までの)新幹線のor『 ひかり』の指定席を取りたいんですが。

(or すいません。JR(or新幹線)のキップ(or指定券)を買いたいんですが。at a JTB office.)

etc., rather than 席を予約したいんですが... (it would make sense but I don't think we really use this phrase when buying a train ticket... I think we say 予約したいんですが when making a reservation at a restaurant, though)

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I did consider simply asking for whatever ticket I wanted

>>> I think you can say like...

「すいません。ハリーポッター、大人2枚。」when buying movie tickets,

「すいません、大人1枚、子ども1枚。」for concert tickets,

「ワンデーパスポート、大人2枚。」at Disneyland,

「すいません、ディズニーランドのチケット買いたいんですが。」for Disneyland tickets at a convenience store, JTB or JR緑の窓口...

etc...

I thought that perhaps I would need to specify that I wished to reserve a seat (rather than getting an unreserved ticket).

>>> When you have a choice and wish to have a reserved ticket/seat then I think you can say like

「指定席で(お願いします)。」「指定券2枚(お願いします)。」

when you'd like an unreserved ticket you can say like

「自由席で(お願いします)。」「自由席(で)2枚(お願いします)。」

(We don't say 自由"券")

But I suppose in any situation where I'd need to reserve a seat, seat reservation would probably be compulsory, so I'd get both tickets as a matter of course.

>>> Yes, in (probably newer) movie theatres and often at classical music concerts I think a seat reservation is compulsory...

Is this something I need to worry about - specifying that I want to reserve a seat, rather than getting an unreserved ticket?

>>> Yes, when you buy a Shinkansen ticket for example, you'd need to specify that. We have:

-自由席 -- unreserved seat (You'd buy 乗車券 and 特急券, but not 指定券)

-指定席 -- reserved seat (指定券=seat reservation ticket; so you'd need 乗車券, 特急券 and 指定券)

-and グリーン車... which is like "First Class" in the UK (but with no free wifi or free refreshments)

I think "すみません。席を予約したいんですが。" is pretty good and not rude at all.

If you want to say it more politely, you can use "すみません。席を予約したいのですが。"